\documentclass[fullpage, 10pt, openany]{book} \usepackage{sectsty, fancyhdr, amsmath} %%%%%%%%%%%Package for writing the answers at the back of the book \usepackage{answers} %%%%%%%%%%Assorted packages \usepackage{makeidx} \usepackage{hangcaption} \usepackage{ntheorem} \usepackage[dvips]{graphicx} %%%%% Postscript drawing packages \usepackage{pstricks, pstricks-add} \usepackage{pst-plot} \usepackage{pst-tree} %%%%%%%%%%%%%%CHAPTER HEADINGS \usepackage[Lenny]{fncychap} \usepackage[colorlinks=true,linkcolor=red, pdftitle={discrete math lecture notes}, pdfauthor={david santos}, bookmarksopen, dvips]{hyperref} \usepackage{pdflscape} \usepackage{thumbpdf} \usepackage[dvips]{graphicx} %%%%% FONTS \usepackage[latin1]{inputenc} \usepackage{t1enc} \usepackage{mathptmx}% instead of package times \usepackage[scaled=0.92]{helvet} % or [scaled=0.92], if you like \usepackage{courier} \usepackage{pifont, stmaryrd} %%for the dingautolists and the proofsymbol \usepackage{marvosym, manfnt, yhmath} \def\arc#1{{\wideparen{#1}}} \usepackage{mathrsfs} %%for the fancy cursive mathscr \usepackage{amsfonts} \usepackage{amssymb} \DeclareSymbolFont{EulerFraktur}{U}{euf}{m}{n} \SetSymbolFont{EulerFraktur}{bold}{U}{euf}{b}{n} \DeclareMathSymbol{!}\mathord {EulerFraktur}{"21} \DeclareMathSymbol{(}\mathopen {EulerFraktur}{"28} \DeclareMathSymbol{)}\mathclose{EulerFraktur}{"29} \DeclareMathSymbol{+}\mathbin {EulerFraktur}{"2B} \DeclareMathSymbol{-}\mathbin {EulerFraktur}{"2D} \DeclareMathSymbol{=}\mathrel {EulerFraktur}{"3D} \DeclareMathSymbol{[}\mathopen {EulerFraktur}{"5B} \DeclareMathSymbol{]}\mathclose{EulerFraktur}{"5D} \DeclareMathSymbol{"}\mathord {EulerFraktur}{"7D} \DeclareMathSymbol{&}\mathord {EulerFraktur}{"26} \DeclareMathSymbol{:}\mathrel {EulerFraktur}{"3A} \DeclareMathSymbol{;}\mathpunct{EulerFraktur}{"3B} \DeclareMathSymbol{?}\mathord {EulerFraktur}{"3F} \DeclareMathSymbol{^}\mathord {EulerFraktur}{"5E} \DeclareMathSymbol{`}\mathord {EulerFraktur}{"12} \DeclareMathDelimiter{(}{EulerFraktur}{"28}{largesymbols}{"00} \DeclareMathDelimiter{)}{EulerFraktur}{"29}{largesymbols}{"01} \DeclareMathDelimiter{[}{EulerFraktur}{"5B}{largesymbols}{"02} \DeclareMathDelimiter{]}{EulerFraktur}{"5D}{largesymbols}{"03} %%%%%%%%%%PAGE FORMATTING \topmargin -.1in \textheight 9in \oddsidemargin -.3in \evensidemargin -.3in \textwidth 7in %%%%%FLOAT PLACEMENT \renewcommand{\topfraction}{.85} \renewcommand{\bottomfraction}{.7} \renewcommand{\textfraction}{.15} \renewcommand{\floatpagefraction}{.66} \renewcommand{\dbltopfraction}{.66} \renewcommand{\dblfloatpagefraction}{.66} \setcounter{topnumber}{9} \setcounter{bottomnumber}{9} \setcounter{totalnumber}{20} \setcounter{dbltopnumber}{9} %%%%Title Page \makeatletter \def\thickhrulefill{\leavevmode \leaders \hrule height 1pt\hfill \kern \z@} \renewcommand{\maketitle}{\begin{titlepage}% \let\footnotesize\small \let\footnoterule\relax \parindent \z@ \reset@font \null\vfil \begin{flushleft} \@title \end{flushleft} \par \hrule height 4pt \par \begin{flushright} \@author \par \end{flushright} \vskip 60\p@ \vspace*{\stretch{2}} \vskip 60\p@ \vspace*{\stretch{2}} \begin{center} \Large\textsf{\today \quad REVISION} \end{center} \end{titlepage}% \setcounter{footnote}{0}% } \makeatother \title{\textcolor{red}{\Large Number Theory for Mathematical Contests }} \author{\textcolor{blue}{David A. 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Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. 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You may include a translation of this License, and all the license notices in the Document, and any Warranty Disclaimers, provided that you also include the original English version of this License and the original versions of those notices and disclaimers. In case of a disagreement between the translation and the original version of this License or a notice or disclaimer, the original version will prevail. If a section in the Document is Entitled ``Acknowledgements'', ``Dedications'', or ``History'', the requirement (section~4) to Preserve its Title (section~1) will typically require changing the actual title. \begin{center} {\Large\bf 9. TERMINATION\par} \phantomsection \addcontentsline{toc}{section}{9. TERMINATION} \end{center} You may not copy, modify, sublicense, or distribute the Document except as expressly provided for under this License. Any other attempt to copy, modify, sublicense or distribute the Document is void, and will automatically terminate your rights under this License. However, parties who have received copies, or rights, from you under this License will not have their licenses terminated so long as such parties remain in full compliance. \begin{center} {\Large\bf 10. FUTURE REVISIONS OF THIS LICENSE\par} \phantomsection \addcontentsline{toc}{section}{10. FUTURE REVISIONS OF THIS LICENSE} \end{center} The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. See http://www.gnu.org/copyleft/. Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License ``or any later version'' applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation. } \QUOTEME{Que a quien robe este libro, o lo tome prestado y no lo devuelva, se le convierta en una serpiente en las manos y lo venza. Que sea golpeado por la par\'{a}lisis y todos sus miembros arruinados. Que languidezca de dolor gritando por piedad, y que no haya coto a su agon\'{\i}a hasta la \'{u}ltima disoluci\'{o}n. Que las polillas roan sus entra\~{n}as y, cuando llegue al final de su castigo, que arda en las llamas del Infierno para siempre.} {Maldici\'{o}n an\'{o}nima contra los ladrones de libros en el monasterio de San Pedro, Barcelona.} \clearpage \twocoltoc{} \chapter*{Preface} \addcontentsline{toc}{chapter}{Preface} \markright{Preface} These notes started in the summer of 1993 when I was teaching {Number Theory} at the Center for Talented Youth Summer Program at the Johns Hopkins University. The pupils were between 13 and 16 years of age. \bigskip The purpose of the course was to familiarise the pupils with contest-type problem solving. Thus the majority of the problems are taken from well-known competitions: \\ \begin{tabular}{ll} {\sc AHSME} & American High School Mathematics Examination \\ {\sc AIME} & American Invitational Mathematics Examination \\ {\sc USAMO} & United States Mathematical Olympiad \\ {\sc IMO} & International Mathematical Olympiad \\ {\sc ITT} & International Tournament of Towns \\ {\sc MMPC} & Michigan Mathematics Prize Competition \\ {\sc $(UM)^2$} & University of Michigan Mathematics Competition \\ {\sc Stanford} & Stanford Mathematics Competition \\ {\sc Mandelbrot} & Mandelbrot Competition \\ \end{tabular} \bigskip Firstly, I would like to thank the pioneers in that course: Samuel Chong, Nikhil Garg, Matthew Harris, Ryan Hoegg, Masha Sapper, Andrew Trister, Nathaniel Wise and Andrew Wong. I would also like to thank the victims of the summer 1994: Karen Acquista, Howard Bernstein, Geoffrey Cook, Hobart Lee, Nathan Lutchansky, David Ripley, Eduardo Rozo, and Victor Yang. \bigskip I would like to thank Eric Friedman for helping me with the typing, and Carlos Murillo for proofreading the notes. \bigskip Due to time constraints, these notes are rather sketchy. Most of the motivation was done in the classroom, in the notes I presented a rather terse account of the solutions. I hope some day to be able to give more coherence to these notes. No theme requires the knowledge of Calculus here, but some of the solutions given use it here and there. The reader not knowing Calculus can skip these problems. Since the material is geared to High School students (talented ones, though) I assume very little mathematical knowledge beyond Algebra and Trigonometry. Here and there some of the problems might use certain properties of the complex numbers. \bigskip A note on the topic selection. I tried to cover most Number Theory that is useful in contests. I also wrote notes (which I have not transcribed) dealing with primitive roots, quadratic reciprocity, diophantine equations, and the geometry of numbers. I shall finish writing them when laziness leaves my weary soul. \bigskip I would be very glad to hear any comments, and please forward me any corrections or remarks on the material herein. \bigskip \hfill \begin{tabular}{l}David A. SANTOS \\ \href{mailto:dsantos@ccp.edu}{dsantos@ccp.edu} \end{tabular} \end{frontmatter} \chapter{Preliminaries} \section{Introduction} \pagenumbering{arabic}\setcounter{page}{1} We can say that no history of mankind would ever be complete without a history of Mathematics. For ages numbers have fascinated Man, who has been drawn to them either for their utility at solving practical problems (like those of measuring, counting sheep, etc.) or as a fountain of solace. Number Theory is one of the oldest and most beautiful branches of Mathematics. It abounds in problems that yet simple to state, are very hard to solve. Some number-theoretic problems that are yet unsolved are: \begin{enumerate} \item (Goldbach's Conjecture) Is every even integer greater than 2 the sum of distinct primes? \item (Twin Prime Problem) Are there infinitely many primes $p$ such that $p + 2$ is also a prime? \item Are there infinitely many primes that are 1 more than the square of an integer? \item Is there always a prime between two consecutive squares of integers? \end{enumerate} In this chapter we cover some preliminary tools we need before embarking into the core of Number Theory. \section{Well-Ordering} The set $ \BBN = \{0, 1, 2, 3, 4, \ldots \}$ of natural numbers is endowed with two operations, addition and multiplication, that satisfy the following properties for natural numbers $a, b,$ and $c$: \begin{enumerate} \item {\bf Closure:} $a + b$ and $ab$ are also natural numbers. \item {\bf Associative laws:} $(a + b) + c = a + (b + c)$ and $a(bc) = (ab)c$. \item {\bf Distributive law:} $a(b + c) = ab + ac.$\item {\bf Additive Identity:} $0 + a = a + 0 = a$ \item {\bf Multiplicative Identity:} $1a = a1 = a$. \end{enumerate} One further property of the natural numbers is the following. \begin{axi}[Well-Ordering Axiom] \label{axi:well_ordering}Every non-empty subset ${\mathscr S}$ of the natural numbers has a least element. \end{axi} As an example of the use of the Well-Ordering Axiom\index{Well-Ordering Axiom}, let us prove that there is no integer between 0 and 1. \begin{exa} Prove that there is no integer in the interval $]0; 1[.$\end{exa} Solution: Assume to the contrary that the set ${\mathscr S}$ of integers in $]0; 1[$ is non-empty. Being a set of positive integers, it must contain a least element, say $m$. Now, $0 < m^2 < m < 1,$ and so $m^2 \in {\mathscr S}$. But this is saying that ${\mathscr S}$ has a positive integer $m^2$ which is smaller than its least positive integer $m$. This is a contradiction and so ${\mathscr S} = \varnothing$. We denote the set of all integers by $\BBZ$, i.e., $$ \BBZ = \{ \ldots -3, -2, -1, 0, 1, 2, 3, \ldots \}.$$ A {\rm rational number} is a number which can be expressed as the ratio $\dfrac{a}{b}$ of two integers $a, b$, where $b \neq 0.$ We denote the set of rational numbers by $\BBQ$. An {\bf irrational number} is a number which cannot be expressed as the ratio of two integers. Let us give an example of an irrational number. \begin{exa} Prove that $\sqrt{2}$ is irrational. \end{exa} Solution: The proof is by contradiction. Suppose that $\sqrt{2}$ were rational, i.e., that $\sqrt{2} = \dfrac{a}{b}$ for some integers $a, b.$ This implies that the set $${\mathscr A} = \{n\sqrt{2}:\ {\rm both} \ n \ {\rm and} \ n\sqrt{2} \ {\rm positive\ integers}\} $$ is nonempty since it contains $a.$ By Well-Ordering ${\mathscr A}$ has a smallest element, say $j = k\sqrt{2}.$ As $\sqrt{2} - 1 > 0$, $$j(\sqrt{2} - 1) = j\sqrt{2} - k\sqrt{2} = (j - k)\sqrt{2}$$ is a positive integer. Since $2 < 2\sqrt{2}$ implies $2 - \sqrt{2} < \sqrt{2}$ and also $j\sqrt{2} = 2k$, we see that $$(j - k)\sqrt{2} = k(2 - \sqrt{2}) < k(\sqrt{2}) = j.$$Thus $(j - k)\sqrt{2}$ is a positive integer in ${\mathscr A}$ which is smaller than $j$. This contradicts the choice of $j$ as the smallest integer in ${\mathscr A}$ and hence, finishes the proof. \begin{exa} Let $a, b, c$ be integers such that $a^6 + 2b^6 = 4c^6 .$ Show that $a = b = c = 0$. \end{exa} Solution: Clearly we can restrict ourselves to nonnegative numbers. Choose a triplet of nonnegative integers $a, b, c$ satisfying this equation and with $$\max (a, b, c) > 0$$ as small as possible. If $a^6 + 2b^6 = 4c^6$ then $a$ must be even, $a = 2a_1$. This leads to $32a_1 ^6 + b^6 = 2c^6.$ Hence $b = 2b_1$ and so $16a_1 ^6 + 32b_1 ^6 = c^6.$ This gives $c = 2c_1,$ and so $a_1 ^6 + 2b_1 ^6 = 4c_1 ^6$. But clearly $\max (a_1, b_1, c_1) < \max (a, b, c).$ This means that all of these must be zero. \begin{exa}[IMO 1988] If $a, b$ are positive integers such that $\displaystyle{\dfrac{a^2 + b^2}{1 + ab}}$ is an integer, then $\displaystyle{\dfrac{a^2 + b^2}{1 + ab}}$ is a perfect square. \end{exa} Solution: Suppose that $\displaystyle{\dfrac{a^2 + b^2}{1 + ab}} = k$ is a counterexample of an integer which is not a perfect square, with $\max (a, b)$ as small as possible. We may assume without loss of generality that $a < b$ for if $a = b$ then $$ 0 < k = \dfrac{2a^2}{a^2 + 1} < 2,$$which forces $k = 1,$ a perfect square. Now, $a^2 + b^2 - k(ab + 1) = 0$ is a quadratic in $b$ with sum of the roots $ka$ and product of the roots $a^2 - k.$ Let $b_1, b$ be its roots, so $b_1 + b = ka$ and $b_1b = a^2 - k.$ As $a, k$ are positive integers, supposing $b_1 < 0$ is incompatible with $a^2 + b_1 ^2 = k(ab_1 + 1).$ As $k$ is not a perfect square, supposing $b_1 = 0$ is incompatible with $a^2 + 0^2 = k(0\cdot a + 1).$ Also $$ b_1 = \dfrac{a^2 - k}{b} < \dfrac{b^2 - k}{b} < b.$$Thus we have found another positive integer $b_1$ for which $\displaystyle{\dfrac{a^2 + b_1 ^2}{1 + ab_1}} = k$ and which is smaller than the smallest $\max (a, b)$. This is a contradiction. It must be the case, then, that $k$ is a perfect square. \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro} Find all integer solutions of $a^3 + 2b^3 = 4c^3.$ \end{pro} \begin{pro} Prove that the equality $x^2 + y^2 + z^2 = 2xyz$ can hold for whole numbers $x, y, z$ only when $x = y = z = 0.$ \end{pro} \end{multicols} \section{Mathematical Induction} The Principle of Mathematical Induction\index{Mathematical Induction} is based on the following fairly intuitive observation. Suppose that we are to perform a task that involves a certain number of steps. Suppose that these steps must be followed in strict numerical order. Finally, suppose that we know how to perform the $n$-th task provided we have accomplished the $n - 1$-th task. Thus if we are ever able to start the job (that is, if we have a base case), then we should be able to finish it (because starting with the base case we go to the next case, and then to the case following that, etc.). Thus in the Principle of Mathematical Induction, we try to verify that some assertion $P(n)$ concerning natural numbers is true for some base case $k_0$ (usually $k_0 = 1,$ but one of the examples below shows that we may take, say $k_0 = 33.$) Then we try to settle whether information on $P(n - 1)$ leads to favourable information on $P(n).$ We will now derive the Principle of Mathematical Induction from the Well-Ordering Axiom. \begin{thm}[Principle of Mathematical Induction]\label{thm:induction} If a set${\mathscr S}$ of non-negative integers contains the integer $0$, and also contains the integer $n + 1$ whenever it contains the integer $n$, then ${\mathscr S} = \BBN$. \end{thm} \begin{pf} Assume this is not the case and so, by the Well-Ordering Principle there exists a least positive integer $k$ not in ${\mathscr S}.$ Observe that $k > 0,$ since $0 \in S$ and there is no positive integer smaller than $0$. As $k - 1 < k,$ we see that $k - 1 \in {\mathscr S}.$ But by assumption $k - 1 + 1$ is also in ${\mathscr S}$, since the successor of each element in the set is also in the set. Hence $k = k - 1 + 1$ is also in the set, a contradiction. Thus ${\mathscr S} = \BBN.$ \end{pf} The following versions of the Principle of Mathematical Induction should now be obvious. \begin{cor} If a set ${\mathscr A}$ of positive integers contains the integer $m$ and also contains $n + 1$ whenever it contains $n$, where $n > m$, then ${\mathscr A}$ contains all the positive integers greater than or equal to $m$.\end{cor} \begin{cor}[Principle of Strong Mathematical Induction] If a set ${\mathscr A}$ of positive integers contains the integer $m$ and also contains $n + 1$ whenever it contains $m + 1, m + 2, \ldots , n$, where $n > m$, then ${\mathscr A}$ contains all the positive integers greater than or equal to $m$.\end{cor} We shall now give some examples of the use of induction. \begin{exa} Prove that the expression $$ 3^{3n + 3} - 26n - 27$$is a multiple of $169$ for all natural numbers $n$. \end{exa}Solution: For $n = 1$ we are asserting that $3^{6} - 53 = 676 = 169\cdot 4$ is divisible by 169, which is evident. Assume the assertion is true for $n - 1, n > 1,$ i.e., assume that $$3^{3n} - 26n - 1 = 169N$$for some integer $N$. Then $$3^{3n + 3} - 26n - 27 = 27\cdot 3^{3n} - 26n - 27 = 27(3^{3n} - 26n - 1) + 676n$$which reduces to $$ 27\cdot 169N + 169\cdot 4n,$$which is divisible by 169. The assertion is thus established by induction. \begin{exa} Prove that $$ (1 + \sqrt{2})^{2n} + (1 - \sqrt{2})^{2n}$$is an even integer and that $$ (1 + \sqrt{2})^{2n} - (1 - \sqrt{2})^{2n} = b\sqrt{2}$$ for some positive integer b, for all integers $n \geq 1.$ \end{exa} Solution: We proceed by induction on $n$. Let $P(n)$ be the proposition: ``$(1 + \sqrt{2})^{2n} + (1 - \sqrt{2})^{2n}$ is even and $(1 + \sqrt{2})^{2n} - (1 - \sqrt{2})^{2n} = b\sqrt{2}$ for some $b \in \BBN$.'' If $n = 1,$ then we see that $$ (1 + \sqrt{2})^2 + (1 - \sqrt{2})^2 = 6,$$an even integer, and $$ (1 + \sqrt{2})^2 - (1 - \sqrt{2})^2 = 4\sqrt{2}.$$ Therefore $P(1)$ is true. Assume that $P(n - 1)$ is true for $n > 1,$ i.e., assume that $$ (1 + \sqrt{2})^{2(n - 1)} + (1 - \sqrt{2})^{2(n - 1)} = 2N$$for some integer $N$ and that $$ (1 + \sqrt{2})^{2(n - 1)} - (1 - \sqrt{2})^{2(n - 1)} = a\sqrt{2}$$for some positive integer $a.$ Consider now the quantity $$(1 + \sqrt{2})^{2n} + (1 - \sqrt{2})^{2n} = (1 + \sqrt{2})^2(1 + \sqrt{2})^{2n - 2} + (1 - \sqrt{2})^2(1 - \sqrt{2})^{2n - 2}.$$This simplifies to $$ (3 + 2\sqrt{2})(1 + \sqrt{2})^{2n - 2} + (3 - 2\sqrt{2})(1 - \sqrt{2})^{2n - 2}.$$Using $P(n - 1)$, the above simplifies to $$ 12N + 2\sqrt{2}a\sqrt{2} = 2(6N + 2a),$$an even integer and similarly $$ (1 + \sqrt{2})^{2n} - (1 - \sqrt{2})^{2n} = 3a\sqrt{2} + 2\sqrt{2}(2N) = (3a + 4N)\sqrt{2}, $$ and so $P(n)$ is true. The assertion is thus established by induction. \begin{exa} Prove that if $k$ is odd, then $2^{n + 2}$ divides $$ k^{2^n} - 1$$ for all natural numbers $n$. \end{exa} Solution: The statement is evident for $n = 1,$ as $k^2 - 1 = (k - 1)(k + 1)$ is divisible by 8 for any odd natural number $k$ because both $(k - 1)$ and $(k + 1)$ are divisible by $2$ and one of them is divisible by $4$. Assume that $2^{n + 2} | k^{2^n} - 1,$ and let us prove that $2^{n + 3} | k^{2^{n + 1}} - 1.$ As $k^{2^{n + 1}} - 1 = (k^{2^n} - 1)(k^{2^n} + 1),$ we see that $2^{n + 2}$ divides $(k^{2n} - 1)$, so the problem reduces to proving that $2 | (k^{2n} + 1).$ This is obviously true since $k^{2n}$ odd makes $k^{2n} + 1$ even. \begin{exa}[USAMO 1978] An integer $n$ will be called {\em good} if we can write $$ n = a_1 + a_2 + \cdots + a_k ,$$ where $a_1 , a_2 , \ldots, a_k$ are positive integers (not necessarily distinct) satisfying $$ \dfrac{1}{a_1} + \dfrac{1}{a_2} + \cdots + \dfrac{1}{a_k} = 1.$$ Given the information that the integers $33$ through $73$ are good, prove that every integer $\geq 33$ is good. \end{exa} Solution: We first prove that if $n$ is good, then $2n + 8$ and $2n + 9$ are good. For assume that $n = a_1 + a_2 + \cdots + a_k$, and $$ 1 = \dfrac{1}{a_1} + \dfrac{1}{a_2} + \cdots + \dfrac{1}{a_k}.$$Then $2n + 8 = 2a_1 + 2a_2 + \cdots + 2a_k + 4 + 4$ and $$ \dfrac{1}{2a_1} + \dfrac{1}{2a_2} + \cdots + \dfrac{1}{2a_k} + \dfrac{1}{4} + \dfrac{1}{4} = \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{4} = 1.$$Also, $2n + 9 = 2a_1 + 2a_2 + \cdots + 2a_k + 3 + 6$ and $$ \dfrac{1}{2a_1} + \dfrac{1}{2a_2} + \cdots + \dfrac{1}{2a_k} + \dfrac{1}{3} + \dfrac{1}{6} = \dfrac{1}{2} + \dfrac{1}{3} +\dfrac{1}{6} = 1.$$ Therefore, \begin{equation} {\rm if} \ n \ {\rm is\ good \ both \ } 2n + 8 \ {\rm and\ } 2n + 9 \ {\rm are\ good.} \ \label{eq:good_integers}\end{equation} We now establish the truth of the assertion of the problem by induction on $n$. Let $P(n)$ be the proposition ``all the integers $n, n + 1, n + 2, \ldots , 2n + 7$'' are good. By the statement of the problem, we see that $P(33)$ is true. But (\ref{eq:good_integers}) implies the truth of $P(n + 1)$ whenever $P(n)$ is true. The assertion is thus proved by induction. We now present a variant of the Principle of Mathematical Induction used by Cauchy to prove the Arithmetic-Mean-Geometric Mean Inequality. It consists in proving a statement first for powers of $2$ and then interpolating between powers of $2$. \begin{thm}[Arithmetic-Mean-Geometric-Mean Inequality] Let $a_1, a_2, \ldots, a_n$ be nonnegative real numbers. Then $$ \sqrt[n]{a_1a_2\cdots a_n} \leq \dfrac{a_1 + a_2 + \cdots + a_n}{n}.$$\end{thm} \begin{pf} Since the square of any real number is nonnegative, we have $$(\sqrt{x_1} - \sqrt{x_2})^2 \geq 0.$$ Upon expanding, \begin{equation}\label{eq:am_gm_2} \dfrac{x_1 + x_2}{2} \geq \sqrt{x_1x_2},\end{equation} which is the Arithmetic-Mean-Geometric-Mean Inequality for $n = 2$. Assume that the Arithmetic-Mean-Geometric-Mean Inequality holds true for $n = 2^{k - 1}, k > 2,$ that is, assume that nonnegative real numbers $w_1, w_2, \ldots , w_{2^{k - 1}}$ satisfy \begin{equation} \label{eq:am_gm_2_kminus1}\dfrac{w_1 + w_2 + \cdots + w_{2^{k - 1}}}{2^{k - 1}} \geq (w_1w_2\cdots w_{2^{k - 1}})^{1/2^{k - 1}}.\end{equation} Using (\ref{eq:am_gm_2}) with $$x_1 = \dfrac{y_1 + y_2 + \cdots + y_{2^{k - 1}}}{2^{k - 1}}$$ and $$x_2 = \dfrac{y_{2^{k - 1} + 1} + \cdots + y_{2^k}}{2^{k - 1}},$$we obtain that $$\small \dfrac{\dfrac{y_1 + y_2 + \cdots + y_{2^{k - 1}}}{2^{k - 1}} + \dfrac{y_{2^{k - 1} + 1} + \cdots + y_{2^k}}{2^{k - 1}}}{2} \geq \quad \left( (\dfrac{y_1 + y_2+ \cdots + y_{2^{k - 1}}}{2^{k - 1}})(\dfrac{y_{2^{k - 1} + 1}+ \cdots + y_{2^k}}{2^{k - 1}})\right)^{1/2}. $$ Applying (\ref{eq:am_gm_2_kminus1}) to both factors on the right hand side of the above , we obtain \begin{equation}\label{eq:am_gm_2_k}\dfrac{y_1 + y_2 + \cdots + y_{2^k}}{2^k} \geq \left( y_1y_2\cdots y_{2^k}\right)^{1/2^k}.\end{equation} This means that the $2^{k - 1}$-th step implies the $2^k$-th step, and so we have proved the Arithmetic-Mean-Geometric-Mean Inequality for powers of 2. Now, assume that $2^{k - 1} < n < 2^{k}$. Let $$ y_1 = a_1, y_2 = a_2, \ldots , y_{n} = a_n, $$ and $$ y_{n + 1} = y_{n + 2} = \cdots = y_{2^k} = \dfrac{a_1 + a_2 + \cdots + a_n}{n}.$$ Let $$ A = \dfrac{a_1 + \cdots + a_n}{n} \ {\rm and} \ G = (a_1\cdots a_n)^{1/n}.$$Using (\ref{eq:am_gm_2_k}) we obtain $$\begin{array}{lll}\dfrac{a_1 + a_2 + \cdots + a_n + (2^k - n)\dfrac{a_1 + \cdots + a_n}{n}}{2^k} & \geq & \\ & & \quad \left( a_1a_2\cdots a_n(\dfrac{a_1 + \cdots + a_n}{n})^{(2^k - n)}\right) ^{1/2^k}, \end{array}$$ which is to say that $$ \dfrac{nA + (2^k - n)A}{2^k} \geq (G^nA^{2^k - n})^{1/2^k}.$$ This translates into $A \geq G$ or $$ \left({a_1a_2\cdots a_n}\right) ^{1/n} \leq \dfrac{a_1 + a_2 + \cdots + a_n}{n},$$ which is what we wanted.\end{pf} \begin{exa} Let $s$ be a positive integer. Prove that every interval $[s; 2s]$ contains a power of $2.$ \end{exa} Solution: If $s$ is a power of $2$, then there is nothing to prove. If $s$ is not a power of 2 then it must lie between two consecutive powers of $2$, i.e., there is an integer $r$ for which $2^r < s <2^{r + 1}$. This yields $2^{r + 1} < 2s$. Hence $s < 2^{r + 1} < 2s,$ which gives the required result. \begin{exa} Let ${\mathscr M}$ be a nonempty set of positive integers such that $4x$ and $[\sqrt{x}]$ both belong to ${\mathscr M}$ whenever $x$ does. Prove that ${\mathscr M}$ is the set of all natural numbers. \end{exa} Solution: We will prove this by induction. First we will prove that $1$ belongs to the set, secondly we will prove that every power of $2$ is in the set and finally we will prove that non-powers of $2$ are also in the set. Since ${\mathscr M}$ is a nonempty set of positive integers, it has a least element, say $a.$ By assumption $\floor{\sqrt{a}}$ also belongs to ${\mathscr M}$, but $\sqrt{a} < a$ unless $a = 1.$ This means that $1$ belongs to ${\mathscr M}.$ \bigskip Since $1$ belongs to ${\mathscr M}$ so does $4$, since $4$ belongs to ${\mathscr M}$ so does $4\cdot 4 = 4^2$, etc.. In this way we obtain that all numbers of the form $4^n = 2^{2n}, n = 1, 2, \ldots$ belong to ${\mathscr M}$. Thus all the powers of $2$ raised to an even power belong to ${\mathscr M}$. Since the square roots belong as well to ${\mathscr M}$ we get that all the powers of $2$ raised to an odd power also belong to ${\mathscr M}$. In conclusion, all powers of $2$ belong to ${\mathscr M}$. \bigskip Assume now that $n\in \BBN$ fails to belong to ${\mathscr M}$. Observe that $n$ cannot be a power of $2$. Since $n \not\in M$ we deduce that no integer in $A_1 = [n^2, (n + 1)^2)$ belongs to ${\mathscr M}$, because every member of $y \in A_1$ satisfies $ [\sqrt{y}] = n$. Similarly no member $z\in A_2 = [n^4, (n + 1)^4)$ belongs to ${\mathscr M}$ since this would entail that $z$ would belong to $A_1$, a contradiction. By induction we can show that no member in the interval $A_r = [n^{2^r}, (n + 1)^{2^r})$ belongs to ${\mathscr M}$. \bigskip We will now show that eventually these intervals are so large that they contain a power of $2$, thereby obtaining a contradiction to the hypothesis that no element of the $A_r$ belonged to ${\mathscr M}$. The function $$\fun{f}{x}{\log _2 x}{\BBR_+ ^*}{\BBR}$$ is increasing and hence $\log _2 (n + 1) - \log _2 n > 0$. Since the function $$\fun{f}{x}{2^{-x}}{\BBR}{\BBR_+ ^*}$$ is decreasing, for a sufficiently large positive integer $k$ we have $$2^{-k} < \log _2 (n + 1) - \log _2 n.$$ This implies that $$(n + 1)^{2^k} > 2n^{2^k}.$$ Thus the interval $[n^{2^k}, 2n^{2^k}]$ is totally contained in $[n^{2^k}, (n + 1)^{2^k})$. But every interval of the form $[s, 2s]$ where $s$ is a positive integer contains a power of $2$. We have thus obtained the desired contradiction. \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro} Prove that $11^{n + 2} + 12^{2n + 1}$ is divisible by $133$ for all natural numbers $n$.\end{pro} \begin{pro} Prove that $$1 - \dfrac{x}{1!} + \dfrac{x(x - 1)}{2!} - \dfrac{x(x - 1)(x - 2)}{3!} \hspace{4mm} \quad $$ $$ \qquad \hspace{4mm} + \cdots + (-1)^n \dfrac{x(x - 1)(x - 2) \cdots (x - n + 1)}{n!}$$equals $$ (-1)^n\dfrac{(x - 1)(x - 2)\cdots (x - n)}{n!} $$for all non-negative integers $n$.\end{pro} \begin{pro} Let $n \in \BBN$. Prove the inequality$$\dfrac{1}{n + 1} + \dfrac{1}{n + 2} + \cdots + \dfrac{1}{3n + 1} > 1.$$\end{pro} \begin{pro} Prove that $$ \underbrace{\sqrt{2 + \sqrt{2 + \cdots + \sqrt{2}}}}_{n \ {\rm radical \ signs}} = 2\cos \dfrac{\pi}{2^{n + 1}}$$for $n \in \BBN$. \end{pro} \begin{pro} Let $a_1 = 3, b_1 = 4,$ and $a_n = 3^{a_{n - 1}}, b_n = 4^{b_{n - 1}}$ when $n > 1.$ Prove that $a_{1000} > b_{999}.$\end{pro} \begin{pro} Let $n \in \BBN , n > 1$. Prove that $$ \dfrac{1\cdot 3\cdot 5\cdots (2n - 1)}{2\cdot 4\cdot 6\cdots (2n)} < \dfrac{1}{\sqrt{3n + 1}}.$$\end{pro} \begin{pro} Prove that if n is a natural number, then$$1\cdot 2 + 2\cdot 5 + \cdots + n\cdot (3n - 1) = n^2(n + 1).$$\end{pro} \begin{pro}Prove that if n is a natural number, then $$ 1^2 + 3^2 + 5^2 + \cdots + (2n - 1)^2 = \dfrac{n(4n^2 - 1)}{3}. $$ \end{pro} \begin{pro}Prove that $$ \dfrac{4^n}{n + 1} < \dfrac{(2n)!}{(n!)^2}$$for all natural numbers $n > 1$. \end{pro}\begin{pro} Prove that the sum of the cubes of three consecutive positive integers is divisible by $9$. \end{pro} \begin{pro}If $|x| \neq 1, n \in \BBN$ prove that $$ \dfrac{1}{1 + x} + \dfrac{2}{1 + x^2} + \dfrac{4}{1 + x^2} + \dfrac{8}{1 + x^8} + \cdots + \dfrac{2^n}{1 + x^{2^n}}$$ equals $$ \dfrac{1}{x - 1} + \dfrac{2^{n + 1}}{1 - x^{2^{n + 1}}}.$$ \end{pro} \begin{pro} Is it true that for every natural number n the quantity $n^2 + n + 41$ is a prime? Prove or disprove!\end{pro} \begin{pro} Give an example of an assertion which is {\em not} true for any positive integer, yet for which the induction step holds.\end{pro} \begin{pro} Give an example of an assertion which is true for the first two million positive integers but fails for every integer greater than $2000000$. \end{pro} \begin{pro} Prove by induction on $n$ that a set having n elements has exactly $2^n$ subsets.\end{pro} \begin{pro} Prove that if $n$ is a natural number, $$ n^5 /5 + n^4 /2 + n^3 /3 - n/30$$is always an integer. \end{pro} \begin{pro}[Halmos]) Every man in a village knows instantly when another's wife is unfaithful, but never when his own is. Each man is completely intelligent and knows that every other man is. The law of the village demands that when a man can PROVE that his wife has been unfaithful, he must shoot her before sundown the same day. Every man is completely law-abiding. One day the mayor announces that there is at least one unfaithful wife in the village. The mayor always tells the truth, and every man believes him. If in fact there are exactly forty unfaithful wives in the village (but that fact is not known to the men,) what will happen after the mayor's announcement?\end{pro} \begin{pro} \begin{enumerate} \item Let $a_1, a_2, \ldots a_n$ be positive real numbers with $$ a_1\cdot a_2 \cdots a_n = 1.$$ Use induction to prove that $$a_1 + a_2 + \cdots + a_n \geq n,$$ with equality if and only if $a_1 = a_2 = \cdots = a_n = 1.$ \item Use the preceding part to give another proof of the Arithmetic-Mean-Geometric-Mean Inequality. \item Prove that if $n > 1$, then $$ 1\cdot 3 \cdot 5 \cdots (2n - 1) < n^n.$$ \item Prove that if $n > 1$ then $$ n\left( (n + 1)^{1/n} - 1\right) < 1 + \dfrac{1}{2} + \cdots + \dfrac{1}{n} .$$ \item Prove that if $n > 1$ then $$ 1 + \dfrac{1}{2} + \cdots + \dfrac{1}{n} < n\left(1 - \dfrac{1}{(n + 1)^{1/n}} + \dfrac{1}{n + 1}\right) .$$ \item Given that u, v, w are positive, $0 < a \leq 1,$ and that $u + v + w = 1,$ prove that $$ \left( \dfrac{1}{u} - a\right)\left(\dfrac{1}{v} - a\right)\left(\dfrac{1}{w} - a\right) \geq 27 - 27a + 9a^2 - a^3.$$ \item Let $y_1, y_2, \ldots , y_n$ be positive real numbers. Prove the {\em Harmonic-Mean- Geometric-Mean Inequality:} $$ \dfrac{n}{\dfrac{1}{y_1} + \dfrac{1}{y_2} + \cdots + \dfrac{1}{y_n}} \leq \sqrt[n]{y_1y_2\cdots y_n}.$$ \item Let $a_1, \ldots , a_n$ be positive real numbers, all different. Set $s = a_1 + a_2 + \cdots + a_n.$ \begin{enumerate} \item Prove that $$ (n - 1)\sum _{1 \leq r \leq n} \dfrac{1}{s - a_r} < \sum _{1 \leq r \leq n} \dfrac{1}{a_r}.$$ \item Deduce that $$ \dfrac{4n}{s} < s\sum _{1 \leq r \leq n} \dfrac{1}{a_r (s - a_r)} < \dfrac{n}{n - 1}\sum _{1 \leq r \leq n} \dfrac{1}{a_r}.$$ \end{enumerate} \end{enumerate}\end{pro} \begin{pro} Suppose that $x_1, x_2, \ldots , x_n$ are nonnegative real numbers with $$x_1 + x_2 + \cdots + x_n \leq 1/2.$$Prove that $$(1 - x_1)(1 - x_2)\cdots (1 - x_n) \geq 1/2.$$\end{pro} \begin{pro} Given a positive integer $n$ prove that there is a polynomial $T_n$ such that $\cos nx = T_n (\cos x)$ for all real numbers $x$. $T_n$ is called the $n$-th {\em Tchebychev Polynomial}\index{Tchebychev Polynomial}. \end{pro} \begin{pro} Prove that $$ \dfrac{1}{n + 1} + \dfrac{1}{n + 2} + \cdots + \dfrac{1}{2n} > \dfrac{13}{24}$$for all natural numbers $n > 1$.\end{pro} \begin{pro} In how many regions will a sphere be divided by n planes passing through its centre if no three planes pass through one and the same diameter?\end{pro} \begin{pro}[IMO 1977] Let $f, f: \BBN \mapsto \BBN$ be a function satisfying $$ f(n + 1) > f(f(n)) $$for each positive integer $n$. Prove that $f(n) = n$ for each n. \end{pro} \begin{pro} Let $F_0(x) = x, F(x) = 4x(1 - x), F_{n + 1}(x) = F(F_n(x)), n = 0, 1, \ldots .$ Prove that $$\int _0 ^1 F_n(x)\, dx = \dfrac{2^{2n - 1}}{2^{2n} - 1}.$$\end{pro} (Hint: Let $x = \sin ^2 \theta$.) \end{multicols} \section{Fibonacci Numbers} The {\em Fibonacci numbers} $f_n$ are given by the recurrence \begin{equation}f_0 = 0, \ f_1 = 1, \ f_{n + 1} = f_{n - 1} + f_n, \ n \geq 1.\end{equation} Thus the first few Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, \ldots . A number of interesting algebraic identities can be proved using the above recursion. \begin{exa} Prove that $$ f_1 + f_2 + \cdots + f_n = f_{n + 2} - 1.$$\end{exa} Solution: We have $$ \begin{array}{ll} f_1 & = f_3 - f_2 \\ f_2 & = f_4 - f_3 \\ f_3 & = f_5 - f_4 \\ \vdots & \vdots \\ f_n & = f_{n + 2} - f_{n + 1} \end{array}$$Summing both columns, $$ f_1 + f_2 + \cdots + f_n = f_{n + 2} - f_2 = f_{n + 2} - 1,$$ as desired. \begin{exa} Prove that $$ f_1 + f_3 + f_5 + \cdots + f_{2n - 1} = f_{2n}.$$\end{exa} Solution: Observe that $$\begin{array}{lcl} f_1 & = & f_2 - f_0 \\ f_3 & = & f_{4} - f_2 \\ f_5 & = & f_6 - f_4 \\ \vdots & \vdots & \vdots \\ f_{2n - 1} & = & f_{2n} - f_{2n - 2} \end{array}$$Adding columnwise we obtain the desired identity. \begin{exa} Prove that $$ f_1 ^2 + f_2 ^2 + \cdots + f_n ^2 = f_{n}f_{n + 1}.$$\end{exa} Solution: We have$$ f_{n - 1}f_{n + 1} = (f_{n + 1} - f_n)(f_n + f_{n - 1}) = f_{n + 1}f_n - f_n ^2 + f_{n + 1}f_{n - 1} - f_nf_{n - 1}.$$Thus $$ f_{n + 1}f_{n} - f_nf_{n - 1} = f_n ^2, $$ which yields $$ f_1 ^2 + f_2 ^2 + \cdots + f_n ^2 = f_nf_{n + 1}.$$ \begin{thm}[Cassini's Identity] \label{thm:cassini2} $$ f_{n - 1}f_{n + 1} - f_n ^2 = (-1)^n, \ n \geq 1.$$ \end{thm} \begin{pf} Observe that $$ \begin{array}{lcl} f_{n - 1}f_{n + 1} - f_n ^2 & = & (f_n - f_{n - 2})(f_n + f_{n - 1}) - f_n ^2 \\ & = & -f_{n - 2}f_n - f_{n - 1}(f_{n - 2} - f_n) \\ & = & -(f_{n - 2}f_n - f_{n - 1} ^2) \end{array}$$Thus if $v_n = f_{n - 1}f_{n + 1} - f_n ^2$, we have $v_n = -v_{n - 1}$. This yields $v_n = (-1)^{n - 1}v_1$ which is to say $$ f_{n - 1}f_{n + 1} - f_n ^2 = (-1)^{n - 1}(f_0f_2 - f_1 ^2) = (-1)^n.$$ \end{pf} \begin{exa}[IMO 1981] Determine the maximum value of $$ m^2 + n^2, $$ where $m, n$ are positive integers satisfying $m, n \in \{ 1, 2, 3, \ldots , 1981\}$ and $$(n^2 - mn - m^2)^2 = 1.$$ \end{exa} Solution: Call a pair $(n, m)$ {\em admissible} if $m, n \in \{ 1, 2, \ldots , 1981\}$ and $(n^2 - mn - m^2)^2 = 1.$ If $m = 1$, then $(1, 1)$ and $(2, 1)$ are the only admissible pairs. Suppose now that the pair $(n_1, n_2)$ is admissible, with $n_2 > 1.$ As $n_1(n_1 - n_2) = n_2 ^2 \pm 1 > 0,$ we must have $n_1 > n_2.$ Let now $n_3 = n_1 - n_2.$ Then $1 = (n_1 ^2 - n_1n_2 - n_2 ^2)^2 = (n_2 ^2 - n_2n_3 - n_3 ^2)^2$, making $(n_2, n_3)$ also admissible. If $n_3 > 1,$ in the same way we conclude that $n_2 > n_3$ and we can let $n_4 = n_2 - n_3$ making $(n_3, n_4)$ an admissible pair. We have a sequence of positive integers $n_1 > n_2 > \ldots$, which must necessarily terminate. This terminates when $n_k = 1$ for some $k$. Since $(n_{k - 1}, 1)$ is admissible, we must have $n_{k - 1} = 2$. The sequence goes thus $1, 2, 3, 5, 8, \ldots , 987, 1597$, i.e., a truncated Fibonacci sequence. The largest admissible pair is thus (1597, 987) and so the maximum sought is $1597^2 + 987^2$. Let $\tau = \dfrac{1 + \sqrt{5}}{2}$ be the Golden Ratio. Observe that $\tau ^{- 1} = \dfrac{\sqrt{5} - 1}{2}$. The number $\tau$ is a root of the quadratic equation $x^2 = x + 1.$ We now obtain a closed formula for $f_n$. We need the following lemma. \begin{lem} If $x^2 = x + 1, n \geq 2$ then we have $x^n = f_n x + f_{n - 1}$. \end{lem} \begin{pf} We prove this by induction on $n.$ For $n = 2$ the assertion is a triviality. Assume that $n > 2$ and that $x^{n - 1} = f_{n - 1}x + f_{n - 2}.$ Then $$\begin{array}{lcl}x^n & = & x^{n - 1}\cdot x \\ & = & (f_{n - 1}x + f_{n - 2})x \\ & = & f_{n - 1} (x + 1) + f_{n - 2}x \\ & = & (f_{n - 1} + f_{n - 2})x + f_{n - 1} \\ & = & f_nx + f_{n - 1} \end{array}$$ \end{pf} \begin{thm}[Binet's Formula]\label{thm:binet_formula} The n-th Fibonacci number is given by $$f_n = \dfrac{1}{\sqrt{5}}\left( \left(\dfrac{1 + \sqrt{5}}{2}\right) ^n - \left(\dfrac{1 - \sqrt{5}}{2} \right)^n\right)$$$n = 0, 2, \ldots .$ \end{thm} \begin{pf} The roots of the equation $x^2 = x + 1$ are $\tau = \dfrac{1 + \sqrt{5}}{2}$ and $1 - \tau = \dfrac{1 - \sqrt{5}}{2}$. In virtue of the above lemma, $$ \tau ^n = \tau f_n + f_{n - 1}$$ and $$(1 - \tau )^n = (1 - \tau )f_n + f_{n - 1}.$$ Subtracting $$ \tau ^n - (1 - \tau )^n = \sqrt{5}f_n,$$ from where {Binet's Formula} follows.\end{pf} \begin{exa}[Ces\`{a}ro] Prove that $$ \sum _{k = 0} ^n \binom{n}{k}2^kf_k = f_{3n}.$$\end{exa} Solution: Using Binet's Formula, $$ \begin{array}{lcl} \sum _{k = 0} ^n \binom{n}{k}2^kf_k & = & \sum _{k = 0} ^n \binom{n}{k}2^k \dfrac{\tau ^k - (1 - \tau)^k}{\sqrt{5}} \\ & = & \dfrac{1}{\sqrt{5}}\left(\sum _{k = 0} ^n \binom{n}{k}\tau ^k - \sum _{k = 0} ^n \binom{n}{k}2^k(1 - \tau)^k \right) \\ & = & \dfrac{1}{\sqrt{5}}\left( (1 + 2\tau)^n - (1 + 2(1 - \tau))^n\right) . \end{array}$$ As $\tau ^2 = \tau + 1, 1 + 2\tau = \tau ^3$. Similarly $1 + 2(1 - \tau) = (1 - \tau)^3$. Thus $$ \sum _{k = 0} ^n \binom{n}{k}2^kf_k = \dfrac{1}{\sqrt{5}}\left( (\tau)^{3n} + (1 - \tau)^{3n}\right) = f_{3n},$$as wanted. \bigskip The following theorem will be used later. \bigskip \begin{thm} \label{thm:cassini1} If $s \geq 1, t\geq 0$ are integers then $$f_{s + t} = f_{s - 1}f_{t} + f_sf_{t + 1}.$$ \end{thm} \begin{pf} We keep $t$ fixed and prove this by using strong induction on $s$. For $s = 1$ we are asking whether $$f_{t + 1} = f_{0}f_{t} + f_1f_{t + 1}, $$which is trivially true. Assume that $s > 1$ and that $f_{s - k + t} = f_{s - k - 1}f_{t} + f_{s - k}f_{t + 1}$ for all $k$ satisfying $1 \leq k \leq s - 1$. We have $$\begin{array}{llll} f_{s + t} & = & f_{s + t - 1} + f_{s + t - 2} & {\rm by \ the \ Fibonacci \ recursion,} \\ & = & f_{s - 1 + t} + f_{s - 2 + t} & {\rm trivially,} \\ & = & f_{s - 2}f_{t} + f_{s - 1}f_{t + 1} + f_{s - 3}f_t + f_{s - 2}f_{t + 1} & {\rm by\ the\ inductive\ assumption} \\ & = & f_t(f_{s - 2} + f_{s - 3}) + f_{t + 1}(f_{s - 1} + f_{s - 2}) & {\rm rearranging,} \\ & = & f_tf_{s - 1} + f_{t + 1}f_s & {\rm by \ the \ Fibonacci \ recursion.}\end{array}$$ This finishes the proof.\end{pf} \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro} Prove that $$ f_{n + 1}f_n - f_{n - 1}f_{n - 2} = f_{2n - 1}, \ n > 2.$$ \end{pro} \begin{pro} Prove that $$ f_{n + 1} ^2 = 4f_nf_{n - 1} + f_{n - 2} ^2, \ n > 1.$$\end{pro} \begin{pro} Prove that $$ f_1f_2 + f_2f_3 + \cdots + f_{2n - 1}f_{2n} = f_{2n} ^2.$$\end{pro} \begin{pro} Let $N$ be a natural number. Prove that the largest $n$ such that $f_n \leq N$ is given by $$ n = \floor{ \dfrac{\log \left(N + \dfrac{1}{2}\right)\sqrt{5}}{\log \left( \dfrac{1 + \sqrt{5}}{2}\right)} }.$$\end{pro} \begin{pro} Prove that $f_n ^2 + f_{n - 1} ^2 = f_{2n + 1}$. \end{pro} \begin{pro} Prove that if $n > 1,$ $$ f_n ^2 - f_{n + l}f_{n - l} = (-1)^{n + l}f_l ^2.$$\end{pro} \begin{pro} Prove that $$ \sum _{k = 1} ^n f_{2k} = \sum _{k = 0} ^n (n - k)f_{2k + 1}.$$\end{pro} \begin{pro} Prove that $$ \sum _{n = 2} ^\infty \dfrac{1}{f_{n - 1}f_{n + 1}} = 1.$$\end{pro} Hint: What is$$ \dfrac{1}{f_{n - 1}f_n} - \dfrac{1}{f_nf_{n + 1}} ?$$ \begin{pro} Prove that $$ \sum _{n = 1} ^\infty \dfrac{f_n}{f_{n + 1}f_{n + 2}} = 1.$$\end{pro} \begin{pro} Prove that $$\sum _{n = 0} ^\infty 1/f_{2^n} = 4 - \tau .$$\end{pro} \begin{pro} Prove that $$ \sum _{n = 1} ^\infty \arctan \dfrac{1}{f_{2n + 1}} = \pi /4.$$\end{pro} \begin{pro} Prove that $$ \lim _{n \rightarrow \infty} \dfrac{f_n}{\tau ^n} = \dfrac{1}{\sqrt{5}}.$$\end{pro} \begin{pro}Prove that $$ \lim _{n\rightarrow\infty} \dfrac{f_{n + r}}{f_n} = \tau ^r.$$ \end{pro} \begin{pro} Prove that $$ \sum _{k = 0} ^{n} \dfrac{1}{f_{2^k}} = 2 + \dfrac{f_{2^n - 2}}{f_{2^n}}.$$ Deduce that $$ \sum _{k = 0} ^\infty \dfrac{1}{f_{2^k}} = \dfrac{7 - \sqrt{5}}{2}.$$ \end{pro} \begin{pro}[Ces\`{a}ro] Prove that $$ \sum _{k = 0} ^n \binom{n}{k}f_k = f_{2n}.$$\end{pro} \begin{pro} Prove that $$ \sum _{n = 1} ^{\infty} \dfrac{f_n}{10^n}$$is a rational number.\end{pro} \begin{pro} Find the exact value of $$ \sum _{k = 1} ^{1994} (-1)^k\binom{1995}{k}f_k.$$\end{pro} \begin{pro} Prove the converse of Cassini's Identity: If $k$ and $m$ are integers such that $|m^2 - km - k^2| = 1,$ then there is an integer $n$ such that $k = \pm f_{n}, m = \pm f_{n + 1}$.\end{pro} \end{multicols} \section{Pigeonhole Principle} The Pigeonhole Principle\index{Pigeonhole Principle} states that if $n + 1$ pigeons fly to $n$ holes, there must be a pigeonhole containing at least two pigeons. This apparently trivial principle is very powerful. Let us see some examples. \begin{exa}[Putnam 1978] Let $A$ be any set of twenty integers chosen from the arithmetic progression $1, 4, \ldots ,100.$ Prove that there must be two distinct integers in $A$ whose sum is $104$. \end{exa} Solution: We partition the thirty four elements of this progression into nineteen groups $\{1 \} , \{ 52 \} $, $\{ 4, 100\}$ , $\{ 7, 97\} $, $\{ 10, 94\}$, \ldots $\{ 49, 55\}.$ Since we are choosing twenty integers and we have nineteen sets, by the Pigeonhole Principle there must be two integers that belong to one of the pairs, which add to $104$. \begin{exa} Show that amongst any seven distinct positive integers not exceeding $126$, one can find two of them, say $a$ and $b$, which satisfy $$ b < a \leq 2b.$$\end{exa} Solution: Split the numbers $\{ 1, 2, 3, \ldots , 126 \}$ into the six sets $$\{ 1, 2\} , \{ 3, 4, 5, 6\} , \{ 7, 8, \ldots , 13, 14\} , \{ 15, 16, \ldots , 29, 30\}, $$ $$\{ 31, 32, \ldots , 61, 62\} \ {\rm and} \ \{ 63, 64, \ldots ,126\} .$$By the Pigeonhole Principle, two of the seven numbers must lie in one of the six sets, and obviously, any such two will satisfy the stated inequality. \begin{exa} Given any set of ten natural numbers between $1$ and $99$ inclusive, prove that there are two disjoint nonempty subsets of the set with equal sums of their elements. \end{exa} Solution: There are $2^{10} - 1 = 1023$ non-empty subsets that one can form with a given 10-element set. To each of these subsets we associate the sum of its elements. The maximum value that any such sum can achieve is $90 + 91 + \cdots + 99 = 945 < 1023.$ Therefore, there must be at least two different subsets that have the same sum. \begin{exa} No matter which fifty five integers may be selected from $$\{ 1, 2, \ldots , 100\},$$ prove that one must select some two that differ by $10$. \end{exa} Solution: First observe that if we choose $n + 1$ integers from any string of $2n$ consecutive integers, there will always be some two that differ by $n$. This is because we can pair the $2n$ consecutive integers $$ \{ a + 1, a + 2, a + 3, \ldots , a + 2n\}$$ into the $n$ pairs $$ \{ a + 1, a + n + 1\}, \{ a + 2, a + n + 2\}, \ldots , \{ a + n, a + 2n\},$$and if $n + 1$ integers are chosen from this, there must be two that belong to the same group. So now group the one hundred integers as follows: $$ \{ 1, 2, \ldots 20 \} , \{ 21, 22, \ldots , 40\} ,$$ $$ \{ 41, 42, \ldots , 60\}, \ \{ 61, 62, \ldots , 80\} $$ and $$ \{ 81, 82, \ldots , 100\} .$$ If we select fifty five integers, we must perforce choose eleven from some group. From that group, by the above observation (let $n = 10$), there must be two that differ by $10$. \begin{exa}[AHSME 1994] Label one disc ``${\bf 1}$'', two discs ``${\bf 2}$'', three discs ``${\bf 3}$'', \ldots , fifty discs $``{\bf 50}$''. Put these $1 + 2 + 3 + \cdots + 50 = 1275$ labeled discs in a box. Discs are then drawn from the box at random without replacement. What is the minimum number of discs that must me drawn in order to guarantee drawing at least ten discs with the same label? \end{exa} Solution: If we draw all the $1 + 2 + \cdots + 9 = 45$ labelled ``${\bf 1}$'', \ldots , ``${\bf 9}$'' and any nine from each of the discs ``${\bf 10}$'', \ldots , ``${\bf 50}$'', we have drawn $45 + 9\cdot 41 = 414$ discs. The 415-th disc drawn will assure at least ten discs from a label. \begin{exa}[IMO 1964] Seventeen people correspond by mail with one another---each one with all the rest. In their letters only three different topics are discussed. Each pair of correspondents deals with only one of these topics. Prove that there at least three people who write to each other about the same topic. \end{exa} Solution: Choose a particular person of the group, say Charlie. He corresponds with sixteen others. By the Pigeonhole Principle, Charlie must write to at least six of the people of one topic, say topic I. If any pair of these six people corresponds on topic I, then Charlie and this pair do the trick, and we are done. Otherwise, these six correspond amongst themselves only on topics II or III. Choose a particular person from this group of six, say Eric. By the Pigeonhole Principle, there must be three of the five remaining that correspond with Eric in one of the topics, say topic II. If amongst these three there is a pair that corresponds with each other on topic II, then Eric and this pair correspond on topic II, and we are done. Otherwise, these three people only correspond with one another on topic III, and we are done again. \begin{exa} Given any seven distinct real numbers $x_1 , \ldots x_7 $, prove that we can always find two, say $a, b$ with $$ 0 < \dfrac{a - b}{1 + ab} < \dfrac{1}{\sqrt{3}}.$$ \end{exa} Solution: Put $x_k = \tan a_k$ for $a_k$ satisfying $-\dfrac{\pi}{2} < a_k < \dfrac{\pi}{2}.$ Divide the interval $(-\dfrac{\pi}{2}, \dfrac{\pi}{2})$ into six non-overlapping subintervals of equal length. By the Pigeonhole Principle, two of seven points will lie on the same interval, say $a_i < a_j$. Then $0 < a_j - a_i < \dfrac{\pi}{6}$. Since the tangent increases in $(-\pi /2, \pi /2)$, we obtain $$0 < \tan (a_j - a_i ) = \dfrac{\tan a_j - \tan a_i}{1 + \tan a_j \tan a_i } < \tan \dfrac{\pi}{6} = \dfrac{1}{\sqrt 3},$$ as desired. \begin{exa}[Canadian Math Olympiad 1981] Let $a_1 , a_2 , \ldots , a_7$ be nonnegative real numbers with $$ a_1 + a_2 + \ldots + a_7 = 1.$$ If $$M = \max _{1 \leq k \leq 5}\,\, a_k + a_{k + 1} + a_{k + 2},$$ determine the minimum possible value that ${\mathscr M}$ can take as the $a_k$ vary. \end{exa} Solution: Since $a_1 \leq a_1 + a_2 \leq a_1 + a_2 + a_3$ and $a_7 \leq a_6 + a_7 \leq a_5 + a_6 + a_7$ we see that ${\mathscr M}$ also equals $$ \max _{1 \leq k \leq 5} \{ a_1, a_7, a_1 + a_2, a_6 + a_7, a_{k} + a_{k + 1} + a_{k + 2} \}.$$We are thus taking the maximum over nine quantities that sum $3(a_1 + a_2 + \cdots + a_7) = 3.$ These nine quantities then average $3/9 = 1/3$. By the Pigeonhole Principle, one of these is $\geq 1/3,$ i.e. $M \geq 1/3$. If $a_1 = a_1 + a_2 = a_1 + a_2 + a_3 = a_2 + a_3 + a_4 = a_3 + a_4 + a_5 = a_4 + a_5 + a_6 = a_5 + a_6 + a_7 = a_7 = 1/3,$ we obtain the 7-tuple $(a_1, a_2, a_3, a_4, a_5, a_6, a_7) = (1/3, 0, 0, 1/3, 0, 0, 1/3),$ which shows that $M = 1/3.$ \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro}[AHSME 1991] A circular table has exactly sixty chairs around it. There are $N$ people seated at this table in such a way that the next person to be seated must sit next to someone. What is the smallest possible value of $N$? \end{pro} Answer: 20. \begin{pro} Show that if any five points are all in, or on, a square of side $1$, then some pair of them will be at most at distance $\sqrt{2}/2.$\end{pro} \begin{pro}[E\"{o}tv\"{o}s, 1947] Prove that amongst six people in a room there are at least three who know one another, or at least three who do not know one another.\end{pro} \begin{pro} Show that in any sum of non-negative real numbers there is always one number which is at least the average of the numbers and that there is always one member that it is at most the average of the numbers.\end{pro} \begin{pro} We call a set ``sum free'' if no two elements of the set add up to a third element of the set. What is the maximum size of a sum free subset of $\{ 1, 2, \ldots , 2n - 1\}$.\end{pro}Hint: Observe that the set $\{ n + 1, n + 2, \ldots , 2n - 1\}$ of $n + 1$ elements is sum free. Show that any subset with $n + 2$ elements is not sum free. \begin{pro}[MMPC 1992] Suppose that the letters of the English alphabet are listed in an arbitrary order. \begin{enumerate} \item Prove that there must be four consecutive consonants. \item Give a list to show that there need not be five consecutive consonants. \item Suppose that all the letters are arranged in a circle. Prove that there must be five consecutive consonants. \end{enumerate} \end{pro} \begin{pro}[Stanford 1953] Bob has ten pockets and forty four silver dollars. He wants to put his dollars into his pockets so distributed that each pocket contains a different number of dollars. \begin{enumerate}\item Can he do so? \item Generalise the problem, considering $p$ pockets and $n$ dollars. The problem is most interesting when $$ n = \dfrac{(p - 1)(p - 2)}{2}.$$ Why? \end{enumerate} \end{pro} \begin{pro} No matter which fifty five integers may be selected from $$\{ 1, 2, \ldots , 100\} , $$ prove that you must select some two that differ by $9$, some two that differ by $10$, some two that differ by $12$, and some two that differ by $13$, but that you need not have any two that differ by $11$. \end{pro} \begin{pro} Let $mn + 1$ different real numbers be given. Prove that there is either an increasing sequence with at least $n + 1$ members, or a decreasing sequence with at least $m + 1$ members.\end{pro} \begin{pro} If the points of the plane are coloured with three colours, show that there will always exist two points of the same colour which are one unit apart. \end{pro} \begin{pro} Show that if the points of the plane are coloured with two colours, there will always exist an equilateral triangle with all its vertices of the same colour. There is, however, a colouring of the points of the plane with two colours for which no equilateral triangle of side $1$ has all its vertices of the same colour. \end{pro} \begin{pro}Let $r_1, r_2, \ldots , r_n, n > 1$ be real numbers of absolute value not exceeding $1$ and whose sum is $0$. Show that there is a non-empty proper subset whose sum is not more than $2/n$ in size. Give an example in which any subsum has absolute value at least $\dfrac{1}{n - 1}.$\end{pro} \begin{pro} Let $r_1, r_2, \ldots , r_n$ be real numbers in the interval $[0, 1]$. Show that there are numbers $\epsilon _k, 1 \leq k \leq n, \epsilon _k = -1, 0, 1$ not all zero, such that $$ \left|\sum _{k = 1} ^n \epsilon _kr_k\right| \leq \dfrac{n}{2^n} .$$\end{pro} \begin{pro}[USAMO, 1979] Nine mathematicians meet at an international conference and discover that amongst any three of them, at least two speak a common language. If each of the mathematicians can speak at most three languages, prove that there are at least three of the mathematicians who can speak the same language.\end{pro} \begin{pro}[USAMO, 1982] In a party with $1982$ persons, amongst any group of four there is at least one person who knows each of the other three. What is the minimum number of people in the party who know everyone else?\end{pro} \begin{pro}[USAMO, 1985] There are $n$ people at a party. Prove that there are two people such that, of the remaining $n - 2$ people, there are at least $\floor{ n/2 } - 1$ of them, each of whom knows both or else knows neither of the two. Assume that ``knowing'' is a symmetrical relationship.\end{pro} \begin{pro}[USAMO, 1986] During a certain lecture, each of five mathematicians fell asleep exactly twice. For each pair of these mathematicians, there was some moment when both were sleeping simultaneously. Prove that, at some moment, some three were sleeping simultaneously. \end{pro} \begin{pro} Let ${\mathscr P}_n$ be a set of $\floor{ en! } + 1$ points on the plane. Any two distinct points of ${\mathscr P}_n$ are joined by a straight line segment which is then coloured in one of $n$ given colours. Show that at least one monochromatic triangle is formed.\end{pro}(Hint: $e = \sum _{n = 0} ^\infty 1/n!.$) \end{multicols} \chapter{Divisibility} \section{Divisibility} \begin{df} If $a \neq 0, b$ are integers, we say that a {\em divides} b if there is an integer $c$ such that $ac = b.$ We write this as $a|b.$ \end{df} If $a$ does not divide $b$ we write $a\not |b.$ The following properties should be immediate to the reader. \begin{thm}\begin{enumerate} \item If $a, b, c, m, n$ are integers with $c|a, c|b$, then $c|(am + nb)$. \item If $x, y, z$ are integers with $x|y , y|z$ then $x|z$. \end{enumerate} \end{thm} \begin{pf} There are integers $s, t$ with $sc = a, tc = b$. Thus $$am + nb = c(sm + tn), $$giving $c|(am + bn).$ Also, there are integers $u, v$ with $xu = y, yv = z.$ Hence $xuv = z$, giving $x|z$. It should be clear that if $a|b$ and $b \neq 0$ then $1 \leq |a| \leq |b|.$\end{pf} \begin{exa} Find all positive integers $n$ for which $$ n + 1|n^2 + 1.$$ \end{exa} Solution: $n^2 + 1 = n^2 - 1 + 2 = (n - 1)(n + 1) + 2$. This forces $n + 1|2$ and so $n + 1 = 1$ or $n + 1 = 2.$ The choice $n + 1 = 1$ is out since $n \geq 1,$ so that the only such $n$ is $n = 1.$ \begin{exa}If $7|3x + 2$ prove that $7|(15x^2 - 11x - 14.)$.\end{exa} Solution: Observe that $15x^2 - 11x - 14 = (3x + 2)(5x - 7).$ We have $7s = 3x + 2$ for some integer $s$ and so $$15x^2 - 11x - 14 = 7s(5x - 7),$$giving the result. \bigskip Among every two consecutive integers there is an even one, among every three consecutive integers there is one divisible by 3, etc.The following theorem goes further. \begin{thm} The product of $n$ consecutive integers is divisible by $n!$. \end{thm} \begin{pf} Assume first that all the consecutive integers $m + 1, m + 2, \ldots , m + n$ are positive. If this is so, the divisibility by $n!$ follows from the fact that binomial coefficients are integers: $$ \binom{m + n}{n} = \dfrac{(m + n)!}{n!m!} = \dfrac{(m + n)(m + n - 1) \cdots (m + 1)}{n!}.$$ If one of the consecutive integers is 0, then the product of them is 0, and so there is nothing to prove. If all the $n$ consecutive integers are negative, we multiply by $(-1)^n$, and see that the corresponding product is positive, and so we apply the first result.\end{pf} \begin{exa} Prove that $6|n^3 - n$, for all integers $n$. \end{exa} Solution: $n^3 - n = (n - 1)n(n + 1)$ is the product of 3 consecutive integers and hence is divisible by $3! = 6.$ \begin{exa}[Putnam 1966] Let $0 < a_1 < a_2 < \ldots < a_{mn + 1}$ be $mn + 1$ integers. Prove that you can find either $m + 1$ of them no one of which divides any other, or $n + 1$ of them, each dividing the following. \end{exa} Solution: Let, for each $1 \leq k \leq mn + 1, n_k$ denote the length of the longest chain, starting with $a_k$ and each dividing the following one, that can be selected from $a_k , a_{k + 1}, \ldots , a_{mn + 1}$. If no $n_k $ is greater than $n$, then the are at least $m + 1 \ n_k$'s that are the same. However, the integers $a_k$ corresponding to these $n_k$'s cannot divide each other, because $a_k |a_l$ implies that $n_k \geq n_l + 1.$ \begin{thm}\label{thm:dividing_fibonacci} If $k|n$ then $f_k|f_n$.\end{thm} \begin{pf} Letting $s = kn, t = n$ in the identity $f_{s + t} = f_{s - 1}f_t + f_{s}f_{t + 1}$ we obtain $$ f_{(k + 1)n} = f_{kn + n} = f_{n - 1}f_{kn} + f_nf_{kn + 1}.$$It is clear that if $f_n|f_{kn}$ then $f_n|f_{(k + 1)n}$. Since $f_n|f_{n\cdot 1}$, the assertion follows.\end{pf} \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro} Given that $5|(n + 2)$, which of the following are divisible by $5$ $$ n^2 - 4, \ n^2 + 8n + 7, \ n^4 - 1, n^2 - 2n ?$$\end{pro} \begin{pro} Prove that $n^5 - 5n^3 + 4n$ is always divisible by $120$. \end{pro} \begin{pro} Prove that $$ \dfrac{(2m)!(3n)!}{(m!)^2 (n!)^3}$$is always an integer.\end{pro} \begin{pro}Demonstrate that for all integer values $n$, $$ n^9 - 6n^7 + 9n^5 - 4n^3$$ is divisible by $8640$.\end{pro} \begin{pro} Prove that if $n > 4$ is composite, then $n$ divides $(n - 1)!$. \\ (Hint: Consider, separately, the cases when $n$ is and is not a perfect square.)\end{pro} \begin{pro} Prove that there is no prime triplet of the form $p, p + 2, p + 4$, except for $3, 5, 7.$\end{pro} \begin{pro} Prove that for $n \in \BBN ,$ $(n!)!$ is divisible by $n!^{(n - 1)!}$\end{pro} \begin{pro}[AIME 1986] What is the largest positive integer $n$ for which $$ (n + 10)|(n^3 + 100)?$$ \end{pro} (Hint: $x^3 + y^3 = (x + y)(x^2 - xy + y^2)$.) \begin{pro}[Olimp\'{\i}ada matem\'{a}tica espa\~{n}ola, 1985] If $n$ is a positive integer, prove that $(n + 1)(n + 2)\cdots (2n)$ is divisible by $2^n$.\end{pro} \end{multicols} \section{Division Algorithm} \begin{thm}[Division Algorithm] \label{thm:division_algorithm} If $a, b$ are positive integers, then there are unique integers $q, r$ such that $a = bq + r, 0 \leq r < b$. \end{thm} \begin{pf} We use the Well-Ordering Principle. Consider the set ${\mathscr S} = \{ a - bk: k \in \BBZ$ and $a \geq bk\}$. Then ${\mathscr S}$ is a collection of nonnegative integers and ${\mathscr S} \neq \varnothing$ as $a - b\cdot 0 \in {\mathscr S}$. By the Well-Ordering Principle, ${\mathscr S}$ has a least element, say $r$. Now, there must be some $q \in \BBZ$ such that $r = a - bq$ since $r \in {\mathscr S}$. By construction, $r \geq 0.$ Let us prove that $r < b$. For assume that $r \geq b.$ Then $r > r - b = a - bq - b = a - (q + 1)b \geq 0$, since $r - b \geq 0.$ But then $a - (q + 1)b \in {\mathscr S}$ and $a - (q + 1)b < r$ which contradicts the fact that $r$ is the smallest member of ${\mathscr S}$. Thus we must have $0 \leq r < b.$ To show that $r$ and $q$ are unique, assume that $bq_1 + r_1 = a = bq_2 + r_2, 0 \leq r_1 < b, 0 \leq r_2 < b. $ Then $r_2 - r_1 = b(q_1 - q_2)$, that is $b|(r_2 - r_1)$. But $|r_2 - r_1| < b,$ whence $r_2 = r_1$. From this it also follows that $q_1 = q_2.$ This completes the proof. \end{pf} It is quite plain that $q = \floor{ a/b },$ where $\floor{ a/b }$ denotes the integral part of $a/b$. It is important to realise that given an integer $n > 0$, the Division Algorithm makes a partition of all the integers according to their remainder upon division by $n$. For example, every integer lies in one of the families $3k, 3k + 1$ or $3k + 2$ where $k \in \BBZ$. Observe that the family $3k + 2, k \in \BBZ$, is the same as the family $3k - 1, k \in \BBZ$. Thus $$ \BBZ = A \cup B \cup C $$ where $$ A = \{ \ldots , -9, -6, -3, 0 , 3, 6, 9, \ldots \} $$is the family of integers of the form $3k, k \in \BBZ$, $$ B = \{ \ldots -8, -5, -2, 1, 4, 7, \ldots \} $$is the family of integers of the form $3k + 1, k \in \BBZ$ and $$ C = \{ \ldots -7, -4, -1, 2, 5, 8, \ldots \}$$ is the family of integers of the form $3k - 1, k \in \BBZ$. \begin{exa}[AHSME 1976] Let r be the remainder when $1059, 1417$ and $2312$ are divided by $d > 1.$ Find the value of $d - r$. \end{exa} Solution: By the Division Algorithm, $1059 = q_1d + r, 1417 = q_2d + r, 2312 = q_3d + r,$ for some integers $q_1, q_2, q_3.$ From this, $358 = 1417 - 1059 = d(q_2 - q_1), 1253 = 2312 - 1059 = d(q_3 - q_1)$ and $895 = 2312 - 1417 = d(q_3 - q_2)$. Hence $d|358 = 2\cdot 179, d|1253 = 7\cdot 179$ and $7|895 = 5\cdot 179$. Since $d > 1,$ we conclude that $d = 179$. Thus (for example) $1059 = 5 \cdot 179 + 164,$ which means that $r = 164.$ We conclude that $d - r = 179 - 164 = 15.$ \begin{exa} Show that $n^2 + 23$ is divisible by $24$ for infinitely many $n$. \end{exa} Solution: $n^2 + 23 = n^2 - 1 + 24 = (n - 1)(n + 1) + 24.$ If we take $n = 24k \pm 1, k = 0, 1, 2, \ldots ,$ all these values make the expression divisible by 24. \begin{df} A {\em prime} number $p$ is a positive integer greater than $1$ whose only positive divisors are $1$ and $p$. If the integer $n > 1$ is not prime, then we say that it is {\em composite.} \end{df} For example, 2, 3, 5, 7, 11, 13, 17, 19 are prime, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20 are composite. The number 1 is neither a prime nor a composite. \begin{exa} Show that if $p > 3$ is a prime, then $24|(p^2 - 1)$. \end{exa} Solution: By the Division Algorithm, integers come in one of six flavours: $6k, 6k \pm 1, 6k \pm 2$ or $6k + 3$. If $p > 3$ is a prime, then $p$ is of the form $p = 6k \pm 1$ (the other choices are either divisible by 2 or 3). But $(6k \pm 1)^2 - 1 = 36k^2 \pm 12k = 12k(3k - 1).$ Since either $k$ or $3k - 1$ is even, $12k(3k - 1)$ is divisible by 24. \begin{exa} Prove that the square of any integer is of the form $4k$ or $4k + 1$. \end{exa} Solution: By the Division Algorithm, any integer comes in one of two flavours: $2a$ or $2a + 1$. Squaring, $$ (2a)^2 = 4a^2,\,\,\, (2a + 1)^2 = 4(a^2 + a) + 1)$$and so the assertion follows. \begin{exa} Prove that no integer in the sequence $$ 11, 111, 1111, 11111, \ldots$$ is the square of an integer. \end{exa} Solution: The square of any integer is of the form $4k$ or $4k + 1$. All the numbers in this sequence are of the form $4k - 1,$ and so they cannot be the square of any integer. \begin{exa} Show that from any three integers, one can always choose two so that $a^3 b - ab^3$ is divisible by $10$. \end{exa} Solution: It is clear that $a^3 b - ab^3 = ab(a - b)(a + b)$ is always even, no matter which integers are substituted. If one of the three integers is of the form $5k$, then we are done. If not, we are choosing three integers that lie in the residue classes $5k \pm 1$ or $5k \pm 2$. Two of them must lie in one of these two groups, and so there must be two whose sum or whose difference is divisible by 5. The assertion follows. \begin{exa} Prove that if $3|(a^2 + b^2 )$, then $3|a$ and $3|b$\end{exa} Solution: Assume $a = 3k \pm 1$ or $b = 3m \pm 1$. Then $a^2 = 3x + 1, b^2 = 3y + 1$. But then $a^2 + b^2 = 3t + 1$ or $a^2 + b^2 = 3s + 2$, i.e., $3\not |(a^2 + b^2).$ \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro} Prove the following extension of the Division Algorithm: if $a$ and $b \neq 0$ are integers, then there are unique integers $q$ and $r$ such that $a = qb + r, 0 \leq r < |b|.$ \end{pro} \begin{pro}Show that if a and b are positive integers, then there are unique integers q and r, and $\epsilon = \pm 1$ such that $a = qb + \epsilon r, -\dfrac{b}{2} < r \leq \dfrac{b}{2}.$ \end{pro} \begin{pro} Show that the product of two numbers of the form $4k + 3$ is of the form $4k + 1.$\end{pro} \begin{pro} Prove that the square of any odd integer leaves remainder $1$ upon division by $8$.\end{pro} \begin{pro}Demonstrate that there are no three consecutive odd integers such that each is the sum of two squares greater than zero.\end{pro} \begin{pro} Let $n > 1$ be a positive integer. Prove that if one of the numbers $2^n - 1, 2^n + 1$ is prime, then the other is composite. \end{pro} \begin{pro}Prove that there are infinitely many integers $n$ such that $4n^2 + 1$ is divisible by both $13$ and $5$.\end{pro} \begin{pro} Prove that any integer $n > 11$ is the sum of two positive composite numbers.\end{pro} Hint: Think of $n - 6$ if $n$ is even and $n - 9$ if $n$ is odd. \begin{pro} Prove that $3$ never divides $n^2 + 1.$\end{pro} \begin{pro} Show the existence of infinitely many natural numbers $x, y$ such that $x(x + 1)|y(y + 1)$ but $$ x\not |y \,\, {\rm and}\,\, (x + 1)\not |y, $$and also $$ x\not |(y + 1)\,\, {\rm and}\,\, (x + 1) \not |(y + 1).$$ \end{pro} Hint: Try $x = 36k + 14, y = (12k + 5)(18k + 7)$. \end{multicols} \section{Some Algebraic Identities} In this section we present some examples whose solutions depend on the use of some elementary algebraic identities. \begin{exa} Find all the primes of the form $n^3 - 1,$ for integer $n > 1.$ \end{exa} Solution: $n^3 - 1 = (n - 1)(n^2 + n + 1).$ If the expression were prime, since $n^2 + n + 1$ is always greater than $1$, we must have $n - 1 = 1,$ i.e. $n = 2.$ Thus the only such prime is $7$. \begin{exa} Prove that $n^4 + 4$ is a prime only when $n = 1$ for $n \in \BBN$. \end{exa} Solution: Observe that $$ \begin{array}{lcl} n^4 + 4 & = & n^4 + 4n^2 + 4 - 4n^2 \\ & = & (n^2 + 2)^2 - (2n)^2 \\ & = & (n^2 + 2 - 2n)(n^2 + 2 + 2n) \\ & = & ((n - 1)^2 + 1)((n + 1)^2 + 1). \end{array}$$Each factor is greater than 1 for $n > 1,$ and so $n^4 + 4$ cannot be a prime. \begin{exa} Find all integers $n \geq 1$ for which $n^4 + 4^n$ is a prime. \end{exa} Solution: The expression is only prime for $n = 1.$ Clearly one must take $n$ odd. For $n \geq 3$ odd all the numbers below are integers: $$\begin{array}{lcl}n^4 + 2^{2n} & = & n^4 + 2n^2 2^{n} + 2^{2n} - 2n^2 2^n \\ & = & (n^2 + 2^n)^2 - \left( n2^{(n + 1)/2}\right)^2 \\ & = & (n^2 + 2^n + n2^{(n + 1)/2})(n^2 + 2^n - n2^{(n + 1)/2}). \end{array}$$ It is easy to see that if $n \geq 3,$ each factor is greater than $1$, so this number cannot be a prime. \begin{exa} Prove that for all $n \in \BBN$ , $n^2$ divides the quantity $$ (n + 1)^n - 1.$$ \end{exa} Solution: If $n = 1$ this is quite evident. Assume $n > 1.$ By the Binomial Theorem, $$ (n + 1)^n - 1 = \sum _{k = 1} ^n \binom{n}{k}n^k ,$$ and every term is divisible by $n^2$. \begin{exa} Prove that if $p$ is an odd prime and if $$ \dfrac{a}{b} = 1 + 1/2 + \cdots + 1/(p - 1),$$ then $p$ divides $a$. \end{exa} Solution: Arrange the sum as $$ 1 + \dfrac{1}{p - 1} + \dfrac{1}{2} + \dfrac{1}{p - 2} + \cdots + \dfrac{1}{(p - 1)/2} + \dfrac{1}{(p + 1)/2}.$$ After summing consecutive pairs, the numerator of the resulting fractions is $p$. Each term in the denominator is $< p$. Since $p$ is a prime, the $p$ on the numerator will not be thus cancelled out. \begin{exa} Prove that {\fboxsep=.2in $$ \framebox{$ x^n - y^n = (x - y)(x^{n - 1} + x^{n - 2}y + x^{n - 3}y^2 + \cdots + xy^{n - 2} + y^{n - 1}) $} $$} Thus {\bf $x - y$ always divides $x^n - y^n$}. \end{exa} Solution: We may assume that $x \neq y, xy \neq 0,$ the result being otherwise trivial. In that case, the result follows at once from the identity $$ \sum _{k = 0} ^{n - 1} a^k = \dfrac{a^{n} - 1}{a - 1} \ a \neq 1,$$upon letting $a = x/y$ and multiplying through by $y^n$. \\ \begin{rem} Without calculation we see that $8767^{2345} - 8101^{2345}$ is divisible by $666$. \end{rem} \begin{exa}[E\H{o}tv\H{o}s 1899] Show that $$ 2903^n - 803^n - 464^n + 261^n$$ is divisible by $1897$ for all natural numbers $n$. \end{exa} Solution: By the preceding problem, $2903^n - 803^n$ is divisible by $2903 - 803 = 2100 = 7\cdot 300 = $, and $261^n - 464^n$ is divisible by $261 - 464 = -203 = 7\cdot (-29)$. Thus the expression $2903^n - 803^n - 464^n + 261^n$ is divisible by 7. Also, $2903^n - 464^n$ is divisible by $2903 - 464 = 9\cdot 271$ and $261^n - 803^n$ is divisible by $-542 = (-2)271$. Thus the expression is also divisible by 271. Since 7 and 271 have no prime factors in common, we can conclude that the expression is divisible by $7\cdot 271 = 1897.$ \begin{exa} [$(UM)^2 C^4 \, 1987$] Given that $1002004008016032$ has a prime factor $p > 250000,$ find it. \end{exa} Solution: If $a = 10^3 , b = 2$ then $$ 1002004008016032 = a^5 + a^4 b + a^3 b^2 + a^2 b^3 + ab^4 + b^5 = \dfrac{ a^6 - b^6 }{ a - b }. $$ This last expression factorises as $$ {\everymath{\displaystyle} \begin{array}{lcl}\dfrac{a^6 - b^6}{a - b} & = & (a + b)(a^2 + ab + b^2)(a^2 - ab + b^2 ) \\ & = & 1002\cdot 1002004\cdot 998004 \\ & = & 4\cdot 4\cdot 1002\cdot 250501 \cdot k,\end{array} }$$where $k < 250000$. Therefore $p = 250501$. \begin{exa}[Gr\"{u}nert, 1856] If $x, y, z, n$ are natural numbers $n \geq z,$ then the relation $$ x^n + y^n = z^n$$does not hold. \end{exa} Solution: It is clear that if the relation $x^n + y^n = z^n$ holds for natural numbers $x, y, z$ then $x < z$ and $y < z$. By symmetry, we may suppose that $x < y$. So assume that $x^n + y^n = z^n$ and $n \geq z.$ Then $$ z^n - y^n = (z - y)(z^{n - 1} + yz^{n - 2} + \cdots + y^{n - 1}) \geq 1\cdot nx^{n - 1} > x^n ,$$ contrary to the assertion that $x^n + y^n = z^n$. This establishes the assertion. \begin{exa} Prove that for $n$ odd, $$ x^n + y^n = (x + y)(x^{n - 1} - x^{n - 2}y + x^{n - 3}y^2 - + -\cdots + - xy^{n - 2} + y^{n - 1}).$$ Thus {\bf if $n$ is odd, $x + y$ divides $x^n + y^n$}. \end{exa} Solution: This is evident by substituting $-y$ for $y$ in example 1.11 and observing that $(-y)^n = -y^n$ for $n$ odd. \begin{exa} Show that $1001$ divides $$1^{1993} + 2^{1993} + 3^{1993} + \cdots + 1000^{1993}.$$ \end{exa} Solution: Follows at once from the previous problem, since each of $1^{1993} + 1000^{1993}, 2^{1993} + 999^{1993}, \ldots , 500^{1993} + 501^{1993}$ is divisible by 1001. \begin{exa}[S250] Show that for any natural number $n$, there is another natural number $x$ such that each term of the sequence $$ x + 1, x^x + 1, x^{x^x} + 1, \ldots $$ is divisible by $n$. \end{exa} Solution: It suffices to take $x = 2n - 1.$ \begin{exa} Determine infinitely many pairs of integers $(m, n)$ such that ${\mathscr M}$ and $n$ share their prime factors and $(m - 1, n - 1)$ share their prime factors.\end{exa} Solution: Take $m = 2^k - 1, n = (2^k - 1)^2 , k = 2, 3, \ldots$. Then $m, n$ obviously share their prime factors and $m - 1 = 2(2^{k - 1} - 1)$ shares its prime factors with $n - 1 = 2^{k + 1}(2^{k - 1} - 1)$. \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro} Show that the integer $$ \underbrace{1 \ldots 1}_{ 91 \ {\rm ones}} $$ is composite.\end{pro} \begin{pro} Prove that $1^{99} + 2^{99} + 3^{99} + 4^{99}$ is divisible by $5$.\end{pro} \begin{pro}Show that if $|ab| \neq 1$, then $a^4 + 4b^4$ is composite.\end{pro} \begin{pro} Demonstrate that for any natural number $n$, the number $$\underbrace{1 \cdots \cdots 1}_{2n \ 1'{\rm s}} - \underbrace{2\cdots \ 2}_{n \ 2'{\rm s}}$$ is the square of an integer.\end{pro} \begin{pro} Let $0 \leq a < b$.\begin{enumerate} \item Prove that $b^n ((n + 1)a - nb) < a^{n + 1}.$ \item Prove that for $n = 1, 2, \ldots$, $$ \left( 1 + \dfrac{1}{n}\right)^n < \left( 1 + \dfrac{1}{n + 1} \right)^{n + 1} \,\,\, n = 1, 2, \ldots .$$ \item Show that $$ \dfrac{b^{n + 1} - a^{n + 1}}{b - a} > (n + 1)a .$$ \item Show that $$ \left( 1 + \dfrac{1}{n}\right)^{n + 1} > \left( 1 + \dfrac{1}{n + 1}\right)^{n + 2} \,\, n = 1, 2, \ldots . $$\end{enumerate}\end{pro} \begin{pro} If $a, b$ are positive integers, prove that $$(a + 1/2)^n + (b + 1/2)^n$$ is an integer only for finitely many positive integers $n$.\end{pro} \begin{pro} Prove that $100|11^{10} - 1.$\end{pro} \begin{pro} Let $A$ and $B$ be two natural numbers with the same number of digits, $A > B$. Suppose that $A$ and $B$ have more than half of their digits on the sinistral side in common. Prove that $$ A^{1/n} - B^{1/n} < \dfrac{1}{n}$$for all $n = 2, 3, 4, \ldots$.\end{pro} \begin{pro} Demonstrate that every number in the sequence $$ 49, 4489, 444889, 44448889, \ldots , \underbrace{4 \cdots \cdots 4}_{n \ 4'{\rm s}}\underbrace{8\cdots \ 8}_{n - 1 \ 8'{\rm s}}9,$$ is the square of an integer.\end{pro} \begin{pro}[Polish Mathematical Olympiad] Prove that if $n$ is an even natural number, then the number $13^n + 6$ is divisible by $7$.\end{pro} \begin{pro} Find, with proof, the unique square which is the product of four consecutive odd numbers.\end{pro} \begin{pro}Prove that the number $2222^{5555} + 5555^{2222}$ is divisible by $7$. \end{pro} (Hint: Consider $$2222^{5555} + 4^{5555} + 5555^{2222} - 4^{2222} + 4^{2222} - 4^{5555}.)$$ \begin{pro} Prove that if $a^n + 1, 1 < a \in \BBN$, is prime, then $a$ is even and $n$ is a power of $2$. Primes of the form $2^{2^k} + 1$ are called {\em Fermat primes}. \end{pro} \begin{pro} Prove that if $a^n - 1, 1 < a \in \BBN$, is prime, then $a = 2$ and $n$ is a prime. Primes of the form $2^n - 1$ are called {\em Mersenne primes}. \end{pro} \begin{pro}[Putnam, 1989] How many primes amongst the positive integers, written as usual in base-ten are such that their digits are alternating $1$'s and $0$'s, beginning and ending in $1$? \end{pro} \begin{pro} Find the least value achieved by $36^k - 5^k, k = 1, 2, \ldots .$\end{pro} \begin{pro}Find all the primes of the form $n^3 + 1$.\end{pro} \begin{pro} Find a closed formula for the product $$ P = (1 + 2)(1 + 2^2 )(1 + 2^{2^2}) \cdots (1 + 2^{2^n}).$$ Use this to prove that for all positive integers $n$, $2^{2^n} + 1$ divides $$ 2^{2^{2^n} + 1} - 2.$$ \end{pro} \begin{pro} Let $a > 1$ be a real number. Simplify the expression $$ \sqrt{a + 2\sqrt{a - 1}} + \sqrt{a - 2 \sqrt{a - 1}}.$$\end{pro} \begin{pro} Let $a, b, c, d$ be real numbers such that $$a^2 + b^2 + c^2 + d^2 = ab + bc + cd + da.$$Prove that $a = b = c = d$.\end{pro} \begin{pro} Let $a, b, c$ be the lengths of the sides of a triangle. Show that $$ 3(ab + bc + ca) \leq (a + b + c)^2 \leq 4(ab + bc + ca).$$\end{pro} \begin{pro}[ITT, 1994]Let $a, b, c, d$ be complex numbers satisfying $$ a + b + c + d = a^3 + b^3 + c^3 + d^3 = 0.$$Prove that a pair of the $a, b, c, d$ must add up to $0$. \end{pro} \begin{pro} Prove that the product of four consecutive natural numbers is never a perfect square. \end{pro} Hint: What is $(n^2 + n - 1)^2$? \begin{pro} Let $k \geq 2$ be an integer. Show that if $n$ is a positive integer, then $n^k$ can be represented as the sum of $n$ successive odd numbers.\end{pro} \begin{pro}[Catalan] Prove that $$ 1 - \dfrac{1}{2} + \dfrac{1}{3} - \dfrac{1}{4} + \cdots + \dfrac{1}{2n - 1} - \dfrac{1}{2n}$$ equals $$ \dfrac{1}{n + 1} + \dfrac{1}{n + 2} + \cdots + \dfrac{1}{2n}. $$\end{pro} \begin{pro}[IMO, 1979] If $a, b$ are natural numbers such that $$ \dfrac{a}{b} = 1 - \dfrac{1}{2} + \dfrac{1}{3} - \dfrac{1}{4} + \cdots - \dfrac{1}{1318} + \dfrac{1}{1319},$$ prove that $1979|a$.\end{pro} \begin{pro}[Polish Mathematical Olympiad] A {\em triangular number} is one of the form $1 + 2 + \ldots + n, n\in \BBN$. Prove that none of the digits $2, 4, 7, 9$ can be the last digit of a triangular number.\end{pro} \begin{pro}Demonstrate that there are infinitely many square triangular numbers.\end{pro} \begin{pro}[Putnam, 1975] Supposing that an integer $n$ is the sum of two triangular numbers, $$ n = \dfrac{a^2 + a}{2} + \dfrac{b^2 + b}{2},$$write $4n + 1$ as the sum of two squares, $4n + 1 = x^2 + y^2$ where $x$ and $y$ are expressed in terms of $a$ and $b$. Conversely, show that if $4n + 1 = x^2 + y^2 ,$ then $n$ is the sum of two triangular numbers.\end{pro} \begin{pro}[Polish Mathematical Olympiad] Prove that amongst ten successive natural numbers, there are always at least one and at most four numbers that are not divisible by any of the numbers $2, 3, 5, 7.$\end{pro} \begin{pro} Show that if $k$ is odd, $$ 1 + 2 + \cdots + n$$ divides $$ 1^k + 2^k + \cdots + n^k .$$ \end{pro} \begin{pro}Are there five consecutive positive integers such that the sum of the first four, each raised to the fourth power, equals the fifth raised to the fourth power?\end{pro} \end{multicols} \chapter{Congruences. $\BBZ_n$} \section{Congruences} The notation $a \equiv b\mod n$ is due to Gau\ss , and it means that $n|(a - b).$ It also indicates that $a$ and $b$ leave the same remainder upon division by $n$. For example, $-8 \equiv -1 \equiv 6 \equiv 13\mod 7$. Since $n|(a - b)$ implies that $\exists k \in \BBZ$ such that $nk = a - b$, we deduce that $a \equiv b\mod n$ if and only if there is an integer $k$ such that $a = b + nk.$ We start by mentioning some simple properties of congruences. \begin{lem}\label{lem:congruence_properties} Let $a, b, c, d, m \in \BBZ, k \in $ with $a \equiv b\mod m$ and $c \equiv d \mod m$. Then \begin{enumerate} \item $a + c \equiv b + d\mod m$ \item $a - c \equiv b - d\mod m$ \item $ac \equiv bd\mod m$ \item $a^k \equiv b^k\mod m$ \item If $f$ is a polynomial with integral coefficients then $f(a) \equiv f(b)\mod m$. \end{enumerate} \end{lem} \begin{pf} As $a \equiv b \mod m$ and $c \equiv d\mod m$, we can find $k_1, k_2 \in \BBZ$ with $a = b + k_1m$ and $c = d + k_2m$. Thus $a \pm c = b \pm d + m(k_1 \pm k_2)$ and $ac = bd + m(k_2b + k_1d)$. These equalities give (1), (2) and (3). Property (4) follows by successive application of (3), and (5) follows from (4). \end{pf} Congruences $\mod 9$ can sometimes be used to check multiplications. For example $875961\cdot 2753 \neq 2410520633.$ For if this were true then $$(8 + 7 + 5 + 9 + 6 + 1)(2 + 7 + 5 + 3) \equiv 2 + 4 + 1 + 0 + 5 + 2 + 0 + 6 + 3 + 3 \ \mod \ 9.$$ But this says that $0\cdot 8 \equiv 8\mod 9$, which is patently false. \begin{exa} Find the remainder when $6^{1987}$ is divided by $37$. \end{exa} Solution: $6^2 \equiv -1\mod 37$. Thus $6^{1987} \equiv 6\cdot 6^{1986} \equiv 6(6^2 )^{993} \equiv 6(-1)^{993} \equiv -6 \equiv 31\mod 37$. \begin{exa} Prove that $7$ divides $3^{2n + 1} + 2^{n + 2}$ for all natural numbers $n$. \end{exa} Solution: Observe that $3^{2n + 1} \equiv 3\cdot 9^n \equiv 3\cdot 2^n\mod 7$ and $2^{n + 2} \equiv 4\cdot 2^n \mod 7$. Hence $$ 3^{2n + 1} + 2^{n + 2} \equiv 7\cdot 2^n \equiv 0 \ \mod \ 7,$$for all natural numbers $n.$ \begin{exa} Prove the following result of Euler: $641|(2^{32} + 1).$\end{exa} Solution: Observe that $641 = 2^7\cdot 5 + 1 = 2^4 + 5^4.$ Hence $2^7\cdot 5 \equiv -1\mod 641$ and $5^4 \equiv -2^4 \mod 641$. Now, $2^7\cdot 5 \equiv -1\mod 641$ yields $5^4 \cdot 2^{28} = (5\cdot 2^7)^4 \equiv (-1)^4 \equiv 1 \mod 641$. This last congruence and $5^4 \equiv -2^4 \mod 641$ yield $-2^{4}\cdot 2^{28} \equiv 1 \mod 641$, which means that $641|(2^{32} + 1).$ \begin{exa} Find the perfect squares $\mod 13$. \end{exa} Solution: First observe that we only have to square all the numbers up to $6$, because $r^2 \equiv (13 - r)^2 \mod 13$. Squaring the nonnegative integers up to $6$, we obtain $0^2 \equiv 0, 1^2 \equiv 1, 2^2 \equiv 4, 3^2 \equiv 9, 4^2 \equiv 3, 5^2 \equiv 12, 6^2 \equiv 10 \mod 13$. Therefore the perfect squares mod 13 are $0$, $1$, $4$, $9$, $3$, $12$, and $10$. \begin{exa} Prove that there are no integers with $x^2 - 5y^2 = 2.$ \end{exa} Solution: If $x^2 = 2 - 5y^2 ,$ then $x^{2} \equiv 2\mod 5$. But $2$ is not a perfect square $\mod 5$. \begin{exa}Prove that $7|(2222^{5555} + 5555^{2222}).$\end{exa} Solution: $2222 \equiv 3 \mod 7$, $5555 \equiv 4\mod 7$ and $3^5 \equiv 5\mod 7$. Now $2222^{5555} + 5555^{2222} \equiv 3^{5555} + 4^{2222} \equiv (3^5)^{1111} + (4^2)^{1111} \equiv 5^{1111} - 5^{1111} \equiv 0\mod 7$. \begin{exa} Find the units digit of $7^{7^7}$.\end{exa} Solution: We must find $7^{7^7}\mod 10$. Now, $7^2 \equiv - 1\mod 10$, and so $7^3 \equiv 7^2 \cdot 7 \equiv -7 \equiv 3\mod 10$ and $7^4 \equiv (7^2)^2 \equiv 1\mod 10$. Also, $7^2 \equiv 1\mod 4$ and so $7^7 \equiv (7^2)^3 \cdot 7 \equiv 3\mod 4$, which means that there is an integer $t$ such that $7^7 = 3 + 4t.$ Upon assembling all this, $$ 7^{7^7} \equiv 7^{4t + 3} \equiv (7^4)^t \cdot 7^3 \equiv 1^t\cdot 3 \equiv 3 \ \mod \ 10. $$Thus the last digit is $3$. \begin{exa} Prove that every year, including any leap year, has at least one Friday $13$-th. \end{exa} Solution: It is enough to prove that each year has a Sunday the 1st. Now, the first day of a month in each year falls in one of the following days: $$\begin{array}{|l|l|l|} \hline {\rm Month} & {\rm Day\ of\ the\ year} & \mod 7 \\ \hline {\rm January} & 1 & 1 \\ \hline {\rm February} & 32 & 4\ \\ \hline {\rm March} & 60 \ {\rm or}\ 61& 4\ {\rm or}\ 5\ \\ \hline {\rm April} & 91 \ {\rm or}\ 92& 0 \ {\rm or}\ 1 \\ \hline {\rm May} & 121\ {\rm or} 122 & 2\ {\rm or}\ 3\\ \hline {\rm June} & 152\ {\rm or}\ 153& 5\ {\rm or}\ 6\\ \hline {\rm July}& 182 \ {\rm or} 183\ & 0\ {\rm or}\ 1\\ \hline {\rm August}& 213\ {\rm or}\ 214 & 3\ {\rm or}\ 4\\ \hline {\rm September}& 244 \ {\rm or}\ 245& 6\ {\rm or}\ 0 \\ \hline {\rm October}& 274\ {\rm or}\ 275 & 1\ {\rm or}\ 2\\ \hline {\rm November}& 305\ {\rm or}\ 306& 4\ {\rm or}\ 5\\ \hline {\rm December}& 335\ {\rm or}\ 336& 6\ {\rm or}\ 0\\ \hline \end{array}$$ (The above table means that, depending on whether the year is a leap year or not, that March 1st is the 50th or 51st day of the year, etc.) Now, each remainder class modulo 7 is represented in the third column, thus each year, whether leap or not, has at least one Sunday the 1st. \begin{exa} Find infinitely many integers $n$ such that $2^n + 27$ is divisible by $7$.\end{exa} Solution: Observe that $2^1 \equiv 2, 2^2 \equiv 4, 2^3 \equiv 1, 2^4 \equiv 2, 2^5 \equiv 4, 2^6 \equiv 1\mod 7$ and so $2^{3k} \equiv 1\mod 3$ for all positive integers $k$. Hence $2^{3k} + 27 \equiv 1 + 27 \equiv 0\mod 7$ for all positive integers $k$. This produces the infinitely many values sought. \begin{exa} Are there positive integers $x, y$ such that $x^3 = 2^y + 15$?\end{exa} Solution: No. The perfect cubes $\mod 7$ are $0$, $1$, and $6$. Now, every power of $2$ is congruent to $1$, $2$, or $4\mod 7$. Thus $2^y + 15 \equiv 2, 3,$ or $5\mod 7$. This is an impossibility. \begin{exa} Prove that $2^k - 5, k = 0, 1, 2, \ldots$ never leaves remainder $1$ when divided by $7$.\end{exa} Solution: $2^1 \equiv 2, 2^2 \equiv 4, 2^3 \equiv 1\mod 7$, and this cycle of three repeats. Thus $2^k - 5$ can leave only remainders $3$, $4$, or $6$ upon division by $7$. \begin{exa}[AIME, 1994] The increasing sequence $$3, 15, 24, 48, \ldots ,$$ consists of those positive multiples of $3$ that are one less than a perfect square. What is the remainder when the $1994$-th term of the sequence is divided by $1000$? \end{exa} Solution: We want $3|n^2 - 1 = (n - 1)(n + 1)$. Since 3 is prime, this requires $n = 3k + 1$ or $n = 3k - 1, k = 1, 2, 3, \ldots $. The sequence $3k + 1, k = 1, 2, \ldots$ produces the terms $n^2 - 1 = (3k + 1)^2 - 1$ which are the terms at even places of the sequence of $3, 15, 24, 48, \ldots $. The sequence $3k - 1, k = 1, 2, \ldots$ produces the terms $n^2 - 1 = (3k - 1)^2 - 1$ which are the terms at odd places of the sequence $3, 15, 24, 48, \ldots $. We must find the 997th term of the sequence $3k + 1, k = 1, 2, \ldots$. Finally, the term sought is $(3(997) + 1)^2 - 1 \equiv (3(-3) + 1)^2 - 1 \equiv 8^2 - 1 \equiv 63\mod 1000$. The remainder sought is $63$. \begin{exa}[USAMO, 1979] Determine all nonnegative integral solutions $$(n_1 , n_2 , \ldots , n_{14})$$ if any, apart from permutations, of the Diophantine equation $$ n_1 ^4 + n_2 ^4 + \cdots + n_{14} ^4 = 1599.$$ \end{exa} Solution: There are no such solutions. All perfect fourth powers $\mod 16$ are $\equiv 0$ or $1\mod 16$. This means that $$ n_1 ^4 + \cdots + n_{14} ^4$$ can be at most $14\mod 16$. But $1599 \equiv 15\mod 16$. \begin{exa}[Putnam, 1986] What is the units digit of $$ \floor{ \dfrac{10^{20000}}{10^{100} + 3} } ?$$ \end{exa} Solution: Set $a - 3 = 10^{100}$. Then $[(10^{20000})/10^{100} + 3] = [(a - 3)^{200}/a] = [\dfrac{1}{a}\sum _{k = 0} ^{200} \binom{200}{k} a^{200 - k} (-3)^k ] = \sum _{k = 0} ^{199} \binom{200}{k} a^{199 - k} (-3)^{k}$. Since $\sum _{k = 0} ^{200} (-1)^k \binom{200}{k} = 0, (3)^{199}\sum _{k = 0} ^{199} (-1)^k \binom{200}{k} = - 3^{199}.$ As $a \equiv 3\mod 10$, $$\sum _{k = 0} ^{199} \binom{200}{k} a^{199 - k}(-3)^k \equiv 3^{199} \sum _{k = 0} ^{199} (-1)^k \binom{200}{k} \equiv - 3^{199} \equiv 3\mod 10.$$ \begin{exa} Prove that for any $a, b, c \in \BBZ, n \in \BBN , n > 3,$ there is an integer $k$ such that $n\not |(k + a), n\not |(k + b), n\not |(k + c)$. \end{exa} Solution: The integers $a, b, c$ belong to at most three different residue classes $\mod n$. Since $n > 3,$ we have more than three distinct residue classes. Thus there must be a residue class, say $k$ for which $-k \not\equiv a, -k \not\equiv b, -k \not\equiv c, \mod n$. This solves the problem. \begin{exa}[Putnam, 1973] Let $a_1 , a_2 , \ldots , a_{2n + 1}$ be a set of integers such that if any one of them is removed, the remaining ones can be divided into two sets of $n$ integers with equal sums. Prove that $a_1 = a_2 = \ldots = a_{2n + 1}$. \end{exa} Solution: As the sum of the $2n$ integers remaining is always even, no matter which of the $a_k$ be taken, all the $a_k$ must have the same parity. The property stated in the problem is now shared by $a_k /2$ or $(a_k - 1)/2$, depending on whether they are all even, or all odd. Thus they are all congruent $\mod 4$. Continuing in this manner we arrive at the conclusion that the $a_k$ are all congruent $\mod 2^k$ for every $k$, and this may only happen if they are all equal. \begin{exa} Prove that $$ (kn)! \equiv 0 \,\, \mod \,\, \prod _{r = 0} ^{n - 1} (n + r)$$ if $n, k \in \BBN , n \geq k \geq 2.$\end{exa} Solution: $(kn)! = M(n - 1)!n(n + 1) \cdots (2n - 1)$ for some integer $M \geq 1.$ The assertion follows. \begin{exa} Let $$ n!! = n!\left(1/2! - 1/3! + \cdots + (-1)^n/n!\right) .$$ Prove that for all $n \in \BBN , n > 3,$ $$ n!! \equiv n! \,\, \mod \,\, (n - 1).$$\end{exa} Solution: We have $$ \begin{array}{lcl}n! - n!! & = & n(n - 1)(n - 2)!( 1 - 1/2! \\ & & \qquad + \cdots + (-1)^{n - 1}/(n - 1)! + (-1)^n/n!) \\ & = & (n - 1)\left( m + (-1)^{n - 1}n/(n - 1) + (-1)^n/(n - 1)\right) \\ & = & (n - 1)\left( m + (-1)^n\right) ,\end{array}$$where ${\mathscr M}$ is an integer, since $(n - 2)!$ is divisible by $k!, k \leq n - 2.$ \begin{exa} Prove that $$ \sum _{k = 0} ^{6n + 2} \binom{6n + 2}{2k}\, 3^k \equiv 0, 2^{3n + 1}, -2^{3n + 1} \, \mod \,\, 2^{3n + 2}$$when $n$ is of the form $2k, 4k + 3$ or $4k + 1$ respectively.\end{exa} Solution: Using the Binomial Theorem, $$ 2S := 2\sum_{k = 0} ^{3n + 1} \binom{6n + 2}{2k} 3^k = (1 + \sqrt{3})^{6n + 2} + (1 - \sqrt{3})^{6n + 2}.$$Also, if $n$ is odd, with $a = 2 + \sqrt{3}, b = 2 - \sqrt{3},$ $${\everymath{\displaystyle}\begin{array}{lcl} \dfrac{1}{2}(a^{3n + 1} + b^{3n + 1}) & = & \sum _{r = 0} ^{\dfrac{3n + 1}{2}} \binom{3n + 1}{2r} 2^{3n + 1 - 2r}3^r. \\ & \equiv & 3^{(3n + 1)/2} \ \mod \ 4 \\ & \equiv & (-1)^{(n - 1)/2} \ \mod \ 4.\end{array}}$$ As $2S = 2^{3n + 1}(a^{3n + 1} + b^{3n + 1}),$ we have, for odd $n$, $$ S \equiv (-1)^{(n - 1)/2}2^{3n + 1} \ \mod \ 2^{3n + 3}. $$ If $n$ is even, $${\everymath{\displaystyle} \begin{array}{lcl} \dfrac{1}{2}(a^{3n + 1} + b^{3n + 1}) & = & \sum _{2r \leq 3n} \binom{3n + 1}{2r + 1} 2^{2r + 1} 3^{3n - 2r} \\ & \equiv & 2(6n + 1)3^{3n} \ \mod \ 8 \\ & \equiv & 4n + 2 \ \mod \ 8. \end{array} } $$So for even $n, S \equiv 2^{3n + 2}{2n + 1} \ \mod \ 2^{3n + 4}.$ \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro} Find the number of all $n, 1 \leq n \leq 25$ such that $n^2 + 15n + 122$ is divisible by $6.$ \end{pro}(Hint: $n^2 + 15n + 122 \equiv n^2 + 3n + 2 = (n + 1)(n + 2)\mod 6$.) \begin{pro}[AIME 1983] Let $a_n = 6^n + 8^n$. Determine the remainder when $a_{83}$ is divided by $49.$ \end{pro} \begin{pro}{\sc (Polish Mathematical Olympiad)} What digits should be put instead of $x$ and $y$ in $30x0y03$ in order to give a number divisible by $13$?\end{pro} \begin{pro} Prove that if $9|(a^3 + b^3 + c^3)$, then $3|abc$, for integers $a, b, c.$ \end{pro} \begin{pro}Describe all integers $n$ such that $10|n^{10} + 1.$\end{pro} \begin{pro} Prove that if $$ a - b, a^2 - b^2, a^3 - b^3, a^4 - b^4, \ldots$$are all integers, then $a$ and $b$ must also be integers.\end{pro} \begin{pro} Find the last digit of $3^{100}$. \end{pro} \begin{pro}[AHSME 1992] What is the size of the largest subset S of $\{ 1, 2, \ldots , 50\}$ such that no pair of distinct elements of S has a sum divisible by $7$?\end{pro} \begin{pro} Prove that there are no integer solutions to the equation $x^2 - 7y = 3.$ \end{pro} \begin{pro} Prove that if $7|a^2 + b^2$ then $7|a$ and $7|b$. \end{pro} \begin{pro} Prove that there are no integers with $$ 800000007 = x^2 + y^2 + z^2 .$$\end{pro} \begin{pro} Prove that the sum of the decimal digits of a perfect square cannot be equal to $1991$. \end{pro} \begin{pro} Prove that $$7|4^{2^n} + 2^{2^n} + 1$$for all natural numbers n.\end{pro} \begin{pro} Prove that $5$ never divides$$ \sum _{k = 0} ^n 2^{3k}\binom{2n + 1}{2k + 1}.$$\end{pro} \begin{pro} Prove that if $p$ is a prime, $\binom{n}{p} - [\dfrac{n}{p}]$ is divisible by $p$, for all $n \geq p.$ \end{pro} \begin{pro} How many perfect squares are there $\mod 2^n$? \end{pro} \begin{pro} Prove that every non-multiple of $3$ is a perfect power of $2$ $\mod 3^n$. \end{pro} \begin{pro} Find the last two digits of $3^{100}$. \end{pro} \begin{pro}[USAMO, 1986] What is the smallest integer $n > 1$, for which the root-mean-square of the first $n$ positive integers is an integer? \end{pro} {\bf Note.} {\footnotesize The root mean square of $n$ numbers $a_1 , a_2 , \ldots , a_n$ is defined to be $$ \left(\dfrac{a_1 ^2 + a_2 ^2 + \cdots + a_n ^2}{n}\right)^{1/2}.$$} \begin{pro}Find all integers $a, b, c, a > 1$ and all prime numbers $p, q, r$ which satisfy the equation $$ p^a = q^b + r^c $$($a, b, c, p, q, r$ need not necessarily be different).\end{pro} \begin{pro} Show that the number $16$ is a perfect $8$-th power $\mod p$ for any prime $p$. \end{pro} \begin{pro}[IMO, 1975] Let $a_1 , a_2, a_3, \ldots$ be an increasing sequence of positive integers. Prove that for every $s \geq 1$ there are infinitely many $a_m$ that can be written in the form $a_m = xa_s + ya_t$ with positive integers x and y and $t > s.$ \end{pro} \begin{pro} For each integer $n > 1$, prove that $n^n - n^2 + n - 1$ is divisible by $(n - 1)^2.$\end{pro} \begin{pro} Let x and $a_i, i = 0, 1, \ldots , k$ be arbitrary integers. Prove that $$ \sum _{i = 0} ^k a_i(x^2 + 1)^{3i}$$is divisible by $x^2 \pm x + 1$ if and only if $\sum _{i = 0} ^k (-1)^ia_i$ is divisible by $x^2 \pm x + 1$. \end{pro} \begin{pro} [$(UM)^2 C^9 \ 1992$] If $x, y, z$ are positive integers with $$ x^n + y^n = z^n $$ for an odd integer $n \geq 3,$ prove that $z$ cannot be a prime-power. \end{pro} \end{multicols} \section{Divisibility Tests} Working base-ten, we have an ample number of rules of divisibility. The most famous one is perhaps the following. \begin{thm}[Casting-out 9's] A natural number $n$ is divisible by $9$ if and only if the sum of it digits is divisible by $9$. \end{thm} \begin{pf} Let $n = a_{k}10^k + a_{k - 1}10^{k - 1} + \cdots + a_110 + a_0$ be the base-10 expansion of $n$. As $10 \equiv 1\mod 9$, we have $10^j \equiv 1\mod 9$. It follows that $n = a_{k}10^k + \cdots + a_110 + a_0 \equiv a_k + \cdots + a_1 + a_0$, whence the theorem. \end{pf} \begin{exa}[AHSME, 1992] The two-digit integers from $19$ to $92$ are written consecutively in order to form the integer $$ 192021222324 \cdots 89909192. $$ What is the largest power of $3$ that divides this number? \end{exa} Solution: By the casting-out-nines rule, this number is divisible by $9$ if and only if $$ 19 + 20 + 21 + \cdots + 92 = 37^2 \cdot 3$$ is. Therefore, the number is divisible by $3$ but not by $9$. \begin{exa}[IMO, 1975] When $4444^{4444}$ is written in decimal notation, the sum of its digits is $A$. Let $B$ be the sum of the digits of $A$. Find the sum of the digits of $B$. ($A$ and $B$ are written in decimal notation.) \end{exa} Solution: We have $4444 \equiv 7\mod 9$, and hence $4444^3 \equiv 7^3 \equiv 1\mod 9$. Thus $4444^{4444} = 4444^{3(1481)} \cdot 4444 \equiv 1\cdot 7 \equiv 7\mod 9$. Let $C$ be the sum of the digits of $B.$ By the casting-out $9$'s rule, $7 \equiv 4444^{4444} \equiv A \equiv B \equiv C\mod 9$. Now, $ 4444\log _{10} 4444 < 4444\log _{10} 10^4 = 17776.$ This means that $4444^{4444}$ has at most 17776 digits, so the sum of the digits of $4444^{4444}$ is at most $9\cdot 17776 = 159984,$ whence $A \leq 159984.$ Amongst all natural numbers $\leq 159984$ the one with maximal digit sum is 99999, so it follows that $B \leq 45.$ Of all the natural numbers $\leq 45,$ $39$ has the largest digital sum, namely $12$. Thus the sum of the digits of $B$ is at most $12$. But since $C \equiv 7\mod 9$, it follows that $C = 7.$ A criterion for divisibility by 11 can be established similarly. For let $n = a_k 10^k + a_{k - 1}10^{k - 1} + \cdots + a_110 + a_0.$ As $10 \equiv - 1\mod 11$, we have $10^j \equiv (-1)^j\mod 11$. Therefore $n \equiv (-1)^k a_k + (-1)^{k - 1}a_{k - 1} + \cdots - a_1 + a_0\mod 11$, that is, $n$ is divisible by $1$1 if and only if the alternating sum of its digits is divisible by 11. For example, $912282219 \equiv 9 - 1 + 2 - 2 + 8 - 2 + 2 - 1 + 9 \equiv 7\mod 11$ and so $912282219$ is not divisible by $11,$ whereas $8924310064539 \equiv 8 - 9 + 2 - 4 + 3 - 1 + 0 - 0 + 6 - 4 + 4 - 3 + 9 \equiv 0\mod 11$, and so $8924310064539$ is divisible by $11$. \begin{exa}[Putnam, 1952] Let $$ f(x) = \sum _{k = 0} ^n a_k x^{n - k}$$be a polynomial of degree $n$ with integral coefficients. If $a_0 , a_n$ and $f(1)$ are all odd, prove that $f(x) = 0$ has no rational roots. \end{exa} Solution: Suppose that $f(a/b) = 0,$ where $a$ and $b$ are relatively prime integers. Then $0 = b^n f(a/b) = a_0b^n + a_1b^{n - 1}a + \cdots + a_{n - 1}ba^{n - 1} + a_na^n.$ By the relative primality of $a$ and $b$ it follows that $a|a_0, b|a_n$, whence $a$ and $b$ are both odd. Hence $$ a_0b^n + a_ab^{n - 1}a + \cdots + a_{n - 1}ba^{n - 1} + a_na^n \equiv a_0 + a_1 + \cdots + a_n = f(1) \equiv 1 \ \mod \ 2,$$but this contradicts that $a/b$ is a root of $f$. \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro}[AHSME 1991] An $n$-digit integer is {\em cute} if its $n$ digits are an arrangement of the set $\{ 1, 2, \ldots ,n\}$ and its first $k$ digits form an integer that is divisible by $k$ for all $k, 1 \leq k \leq n$. For example, $321$ is a cute three-digit number because $1$ divides $3$, $2$ divides $32$, and $3$ divides $321$. How many cute six-digit integers are there? \end{pro} Answer: $2$. \begin{pro} How many ways are there to roll two distinguishable dice to yield a sum that is divisible by three?\end{pro} Answer: 12. \begin{pro} Prove that a number is divisible by $2^k, k \in \BBN$ if and only if the number formed by its last k digits is divisible by $2^k.$ Test whether $$90908766123456789999872$$ is divisible by $8$.\end{pro} \begin{pro} An old receipt has faded. It reads $88$ chickens at the total of $ \$ x4.2y$, where $x$ and $y$ are unreadable digits. How much did each chicken cost?\end{pro} Answer: $73$ cents. \begin{pro} Five sailors plan to divide a pile of coconuts amongst themselves in the morning. During the night, one of them wakes up and decides to take his share. After throwing a coconut to a monkey to make the division come out even, he takes one fifth of the pile and goes back to sleep. The other four sailors do likewise, one after the other, each throwing a coconut to the monkey and taking one fifth of the remaining pile. In the morning the five sailors throw a coconut to the monkey and divide the remaining coconuts into five equal piles. What is the smallest amount of coconuts that could have been in the original pile? \end{pro}Answer: $15621$ \begin{pro} Prove that a number which consists of $3^n$ identical digits is divisible by $3^n$. For example, $111\ 111 \ 111$ is divisible by $9$.\end{pro} \begin{pro}[$(UM)^2 C^8 \ 1991$] Suppose that $a_0 , a_1 , \ldots a_n$ are integers with $a_n \neq 0$, and let $$ p(x) = a_0 + a_1 x + \cdots + a_n x^n .$$Suppose that $x_0$ is a rational number such that $p(x_0 ) = 0$. Show that if $1 \leq k \leq n$, then $$ a_k x_0 + a_{k + 1} x^2 _0 + \cdots + a_n x^{n - k + 1}$$is an integer. \end{pro} \begin{pro} $1953$ digits are written in a circular order. Prove that if the $1953$-digit numbers obtained when we read these digits in dextrogyral sense beginning with one of the digits is divisible by $27$, then if we read these digits in the same direction beginning with any other digit, the new $1953$-digit number is also divisible by $27$.\end{pro} \begin{pro}[Lagrange] Prove that $$ f_{n + 60} \equiv f_n \ \mod \ 10.$$Thus the last digit of a Fibonacci number recurs in cycles of length $60$.\end{pro} \begin{pro} Prove that $$ f_{2n + 1} \equiv f_{n + 1} ^2 \ \mod \ f_n ^2.$$\end{pro} \end{multicols} \section{Complete Residues} The following concept will play a central role in our study of integers. \begin{df} If $a \equiv b \mod n$ then $b$ is called a {\em residue} of $a$ modulo $n$. A set $a_1, a_2, \ldots a_n$ is called a {\em complete residue system} modulo $n$ if for every integer $b$ there is exactly one index $j$ such that $b \equiv a_j\mod n$. \end{df} It is clear that given any finite set of integers, this set will form a complete set of residues modulo $n$ if and only if the set has $n$ members and every member of the set is incongruent modulo $n$. For example, the set ${\mathscr A} = \{ 0, 1, 2, 3, 4, 5\}$ forms a complete set of residues $\mod 6$, since any integer $x$ is congruent to one and only one member of ${\mathscr A}$. Notice that the set ${\mathscr B} = \{ -40, 6, 7, 15, 22, 35\}$ forms a complete residue set $\mod 6$, but the set ${\mathscr C} = \{ -3, -2, -1, 1, 2, 3\}$ does not, as $-3 \equiv 3\mod 6$. \begin{table}[h] \begin{minipage}{7cm} \centering \begin{tabular}{|c|c|c|c|} \hline $+_3$ & ${\sf{0}}$ & ${\sf{1}}$ & ${\sf{2}}$ \\ \hline ${\sf{0}}$ & ${\sf{0}}$ & ${\sf{1}}$ & ${\sf{2}}$ \\ \hline ${\sf{1}}$ & ${\sf{1}}$ & ${\sf{2}}$ & ${\sf{0}}$ \\ \hline ${\sf{2}}$ & ${\sf{2}}$ & ${\sf{0}}$ & ${\sf{1}}$ \\ \hline \end{tabular} \vspace{2.2cm} \footnotesize\hangcaption{Addition Table for $\BBZ_3$} \end{minipage} \begin{minipage}{7cm} \centering \begin{tabular}{|c|c|c|c|c|c|c|} \hline $+_6$ & ${\sf{0}}$ & ${\sf{1}}$ & ${\sf{2}}$ & ${\sf 3}$ & ${\sf 4}$ & ${\sf 5}$ \\ \hline ${\sf{0}}$ & ${\sf{0}}$ & ${\sf{1}}$ & ${\sf{2}}$ & ${\sf 3}$ & ${\sf 4}$ & ${\sf 5}$ \\ \hline ${\sf{1}}$ & ${\sf{1}}$ & ${\sf{2}}$ & ${\sf{3}}$ & ${\sf 4}$ & ${\sf 5}$ & ${\sf 0}$ \\ \hline ${\sf{2}}$ & ${\sf{2}}$ & ${\sf{3}}$ & ${\sf{4}}$ & ${\sf 5}$ & ${\sf 0}$ & ${\sf 1}$ \\ \hline ${\sf{3}}$ & ${\sf{3}}$ & ${\sf{4}}$ & ${\sf{5}}$ & ${\sf 0}$ & ${\sf 1}$ & ${\sf 2}$ \\ \hline ${\sf{4}}$ & ${\sf{4}}$ & ${\sf{5}}$ & ${\sf{0}}$ & ${\sf 1}$ & ${\sf 2}$ & ${\sf 3}$ \\ \hline ${\sf{5}}$ & ${\sf{5}}$ & ${\sf{0}}$ & ${\sf{1}}$ & ${\sf 2}$ & ${\sf 3}$ & ${\sf 4}$ \\ \hline \end{tabular} \vspace{1cm}\footnotesize\hangcaption{Addition Table for $\BBZ_6$} \end{minipage}\label{table:addition_z_6} \end{table} Tied up with the concept of complete residues is that of $\BBZ_n$. As an example, let us take $n = 3.$ We now let $\sf{0}$ represent all those integers that are divisible by $3$, $\sf{1}$ represent all those integers that leave remainder 1 upon division by 3, and $\sf{2}$ all those integers that leave remainder $2$ upon division by $3$, and consider the set $\BBZ_3 = \{ \sf{0}, \sf{1}, \sf{2}\}$. We define addition in $\BBZ_3$ as follows. Given ${\sf a}, {\sf b} \in \BBZ_3$ we consider $a + b\mod 3$. Now, there is $c \in \{ 0, 1, 2\}$ such that $a + b \equiv c\mod 3$. We then define ${\sf a} +_3 {\sf b}$ to be equal to ${\sf c}.$ Table \ref{table:addition_z_6} contains all the possible additions. We observe that $\BBZ_3$ together with the operation $+_3$ as given in Table \ref{table:addition_z_6} satisfies the following properties: \begin{enumerate} \item The element $\sf{0}\in \BBZ_3$ is an {\em identity element} for $\BBZ_3$, i.e. $\sf{0}$ satisfies $\sf{0} +_3 \sf{a} = \sf{a} +_3 \sf{0} = \sf{a}$ for all $\sf{a}\in\BBZ_3$ \item Every element $\sf{a}\in\BBZ_3$ has an {\em additive inverse} $\sf{b}$, i.e., an element such that $\sf{a} +_3 \sf{b} = \sf{b} +_3 \sf{a} = \sf{0}$. We denote the additive inverse of $\sf{a}$ by $-\sf{a}$. In $\BBZ_3$ we note that $-\sf{0} = \sf{0}, -\sf{1} = \sf{2}, -\sf{2} = \sf{1}$. \item The operation addition in $\BBZ_3$ is {\em associative}, that is, for all $\sf{a}, \sf{b}, \sf{c}\in \BBZ_3$ we have $\sf{a} +_3 (\sf{b} +_3 \sf{c}) = (\sf{a} +_3 \sf{b}) +_3 \sf{c}$. \end{enumerate} We then say that $<\BBZ_3, +_3>$ forms a {\em group} and we call it the {\em group of residues under addition} $\mod 3$. Similarly we define $<\BBZ_n, +_n>$, as the {\em group of residues under addition} $\mod n$. As a further example we present the addition table for $<\BBZ_6, +_6>$ on Table (1.2). We will explore later the multiplicative structure of $\BBZ_n$. \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro} Construct the addition tables for $\BBZ_8$ and $\BBZ_9$.\end{pro} \begin{pro} How many distinct ordered pairs $({\sf a}, {\sf b}) \neq ({\sf 0}, {\sf 0})$ are in $\BBZ_{12}$ such that ${\sf a} +_{12} {\sf b} = {\sf 0}$?\end{pro} \end{multicols} \chapter{Unique Factorisation} \section{GCD and LCM} If $a, b \in \BBZ$, not both zero, the largest positive integer that divides both $a, b$ is called the {\em greatest common divisor of a and b.} This is denoted by $(a, b)$ or sometimes by $\gcd (a, b)$. Thus if $d|a$ and $d|b$ then $d|(a, b)$, because any common divisor of $a$ and $b$ must divide the largest common divisor of $a$ and $b$. For example, $ (68, -6) = 2, \gcd (1998, 1999) = 1.$ If $ (a, b) = 1,$ we say that $a$ and $b$ are {\em relatively prime or coprime.} Thus if $a, b$ are relatively prime, then they have no factor greater than 1 in common. If $a, b$ are integers, not both zero, the smallest positive integer that is a multiple of $a, b$ is called the {\em least common multiple of a and b}. This is denoted by $[a, b]$. We see then that if $a|c$ and if $b|c$, then $[a, b]|c$, since $c$ is a common multiple of both $a$ and $b$, it must be divisible by the smallest common multiple of $a$ and $b$. The most important theorem related to gcd's is probably the following. \begin{thm}[Bachet-Bezout Theorem] \label{thm:bachet_bezout} The greatest common divisor of any two integers $a, b$ can be written as a linear combination of $a$ and $b$, i.e., there are integers $x, y$ with $$ (a, b) = ax + by.$$ \end{thm} \begin{pf} Let ${\mathscr A} = \{ ax + by| ax + by > 0, x, y \in \BBZ\}$. Clearly one of $\pm a, \pm b$ is in ${\mathscr A}$, as both $a, b$ are not zero. By the Well Ordering Principle, ${\mathscr A}$ has a smallest element, say $d$. Therefore, there are $x_0, y_0$ such that $d = ax_0 + by_0.$ We prove that $d = (a, b).$ To do this we prove that $d|a, d|b$ and that if $t|a, t|b,$ then $t|d.$ We first prove that $d|a.$ By the Division Algorithm, we can find integers $q, r, 0 \leq r < d$ such that $a = dq + r.$ Then $$ r = a - dq = a(1 - qx_0) - by_0.$$If $r > 0,$ then $r \in {\mathscr A}$ is smaller than the smaller element of ${\mathscr A}$, namely $d,$ a contradiction. Thus $r = 0.$ This entails $dq = a,$ i.e. $d|a.$ We can similarly prove that $d|b.$ Assume that $t|a, t|b$. Then $a = tm, b = tn$ for integers $m, n.$ Hence $d = ax_0 + bx_0 = t(mx_0 + ny_0),$ that is, $t|d.$ The theorem is thus proved. \end{pf} \begin{rem} It is clear that any linear combination of $a, b$ is divisible by $(a, b)$. \end{rem} \begin{lem}[Euclid's Lemma] \label{lem:euclid} If $a|bc$ and if $ (a, b) = 1$, then $a|c.$ \end{lem} \begin{pf} As $ (a, b) = 1$, by the Bachet-Bezout Theorem, there are integers $x, y$ with $ax + by = 1.$ Since $a|bc,$ there is an integer $s$ with $as = bc.$ Then $c = c\cdot 1 = cax + cby = cax + asy.$ From this it follows that $a|c,$ as wanted.\end{pf} \begin{thm}\label{thm:dividing_by_gcd} If $ (a, b) = d$, then $$ (\dfrac{a}{d}, \dfrac{b}{d}) = 1. $$ \end{thm} \begin{pf} By the Bachet-Bezout Theorem, there are integers $x, y$ such that $ax + by = d.$ But then $(a/d)x + (b/d)y = 1,$ and $a/d, b/d$ are integers. But this is a linear combination of $a/d, b/d$ and so $(a/d, b/d)$ divides this linear combination, i.e., divides 1. We conclude that $ (a/d, b/d) = 1.$\end{pf} \begin{thm}\label{thm:multiplying_by_gcd} Let c be a positive integer. Then $$ (ca, cb) = c (a, b).$$\end{thm} \begin{pf} Let $d_1 = (ca, cb)$ and $d_2 = (a, b).$ We prove that $d_1|cd_2$ and $cd_2|d_1.$ As $d_2|a$ and $d_2|b,$ then $cd_2|ca, cd_2|cb$. Thus $cd_2$ is a common divisor of $ca$ and $cb$ and hence $d_1|cd_2.$ By the Bachet-Bezout Theorem we can find integers $x, y$ with $d_1 = acx + bcy = c(ax + by).$ But $ax + by$ is a linear combination of $a, b$ and so it is divisible by $d_2.$ There is an integer $s$ then such that $sd_2 = ax + by.$ It follows that $d_1 = csd_2,$ i.e., $cd_2|d_1$. \end{pf} \begin{rem} It follows similarly that $ (ca, cb) = |c|(a, b)$ for any non-zero integer $c$. \end{rem} \begin{lem} For nonzero integers a, b, c,$$ (a, bc) = (a, (a, b)c).$$ \end{lem} \begin{pf} Since $ (a, (a, b)c)$ divides $(a, b)c$ it divides $bc$. Thus $\gcd (a, (a, b)c)$ divides $a$ and $bc$ and hence $\gcd (a, (a, b)c)|\gcd (a, bc).$ On the other hand, $(a, bc)$ divides $a$ and $bc$, hence it divides $ac$ and $bc$. Therefore $(a, bc)$ divides $(ac, bc) = c(a, b)$. In conclusion, $(a, bc)$ divides $a$ and $c(a, b)$ and so it divides $(a, (a, b)c)$. This finishes the proof.\end{pf} \begin{thm} $ (a^2, b^2) = (a, b)^2$.\end{thm} \begin{pf} Assume that $(m, n) = 1.$ Using the preceding lemma twice, $$ (m^2, n^2) = (m^2, (m^2, n)n) = (m^2, (n, (m, n)m)n).$$As $ (m, n) = 1,$ this last quantity equals $ (m^2, n).$ Using the preceding problem again, $$ (m^2, n) = (n, (m, n)m) = 1.$$Thus $ (m, n) = 1$ implies $(m^2, n^2) = 1$. By Theorem \ref{thm:dividing_by_gcd}, $$ \left( \dfrac{a}{(a, b)}, \dfrac{b}{(a, b)}\right) = 1,$$and hence $$ \left( \dfrac{a^2}{(a, b)^2}, \dfrac{b^2}{(a, b)^2}\right) = 1.$$By Theorem \ref{thm:multiplying_by_gcd}, upon multiplying by $(a, b)^2$, we deduce $$ (a^2, b^2) = (a, b)^2,$$which is what we wanted.\end{pf} \begin{exa} Let $(a, b) = 1$. Prove that $(a + b, a^2 - ab + b^2) = 1$ or $3$.\end{exa} Solution: Let $d = (a + b, a^2 - ab + b^2)$. Now $d$ divides $$ (a + b)^2 - a^2 + ab - b^2 = 3ab.$$Hence $d$ divides $3b(a + b) - 3ab = 3b^2. $ Similarly, $d|3a^2$. But then $d|(3a^2, 3b^2) = 3(a^2, b^2) = 3(a, b)^2 = 3.$ \begin{exa} Let $a, a \neq 1, m, n$ be positive integers. Prove that $$ (a^m - 1, a^n - 1) = a^{(m, n)} - 1.$$ \end{exa} Solution: Set $d = (m, n), sd = m, td = n$. Then $a^m - 1 = (a^d )^s - 1$ is divisible by $a^d - 1$ and similarly, $a^n - 1$ is divisible by $a^d - 1$. Thus $(a^d - 1)|(a^m - 1, a^n - 1)$. Now, by the Bachet-Bezout Theorem there are integers $x, y$ with $mx + ny = d.$ Notice that $x$ and $y$ must have opposite signs (they cannot obviously be both negative, since then $d$ would be negative. They cannot both be positive because then $d \geq m + n$, when in fact we have $d \leq m, d \leq n$). So, assume without loss of generality that $x > 0, y \leq 0$. Set $t = (a^m - 1, a^n - 1)$. Then $t|(a^{mx} - 1)$ and $t|(a^{-ny} - 1).$ Hence, $t|((a^{mx} - 1) - a^{d}(a^{-ny} - 1)) = a^d - 1.$ The assertion is established. \begin{exa}[IMO, 1959] Prove that the fraction $\dfrac{21n + 4}{14n + 3}$ is irreducible for every natural number $n$. \end{exa} Solution: $2(21n + 4) - 3(14n + 3) = -1.$ Thus the numerator and the denominator have no common factor greater than $1$. \begin{exa}[AIME, 1985] The numbers in the sequence $$ 101, 104, 109, 116, \ldots $$ are of the form $a_n = 100 + n^2 , n = 1, 2, \ldots$. For each $n$ let $d_n = (a_n , a_{n + 1})$. Find $\max _{n \geq 1}\,\, d_n$. \end{exa} Solution: We have the following: $d_n = (100 + n^2 , 100 + (n + 1)^2 ) = (100 + n^2 , 100 + n^2 + 2n + 1) = (100 + n^2 , 2n + 1)$. Thus $d_n |(2(100 + n^2 ) - n (2n + 1)) = 200 - n$. Therefore $d_n |(2(200 - n) + (2n + 1)) = 401$. This means that $d_n |401$ for all $n$. Could it be that large? The answer is yes, for let $n = 200,$ then $a_{200} = 100 + 200^2 = 100(401)$ and $a_{201} = 100 + 201^2 = 40501 = 101(401)$. Thus $\max _{n \geq 1} \, d_n = 401.$ \begin{exa} Prove that if $m$ and $n$ are natural numbers and $m$ is odd, then $(2^m - 1, 2^n + 1) = 1.$\end{exa} Solution: Let $d = (2^m - 1, 2^n + 1).$ It follows that $d$ must be an odd number, and $2^m - 1 = kd, 2^n + 1 = ld,$ for some natural numbers $k, l.$ Therefore, $2^{mn} = (kd + 1)^n = td + 1$, where $t = \sum _{j = 0} ^{n - 1} \binom{n}{j}k^{n - j} d^{n - j - 1}$. In the same manner, $2^{mn} = (ld - 1)^m = ud - 1,$ where we have used the fact that $m$ is odd. As $td + 1 = ud - 1,$ we must have $d|2,$ whence $d = 1.$ \begin{exa} Prove that there are arbitrarily long arithmetic progressions in which the terms are pairwise relatively prime. \end{exa} Solution: The numbers $km! + 1, k = 1, 2, \ldots , m$ form an arithmetic progression of length $m$ and common difference $m!$. Suppose that $d|(lm! + 1), d|(sm! + 1), 1 \leq l < s \leq m$. Then $d|(s(lm! + 1) - l(sm! + 1)) = (s - l) < m.$ Thus $1 \leq d < m$ and so, $d|m!$. But then $d|(sm! + 1 - sm!) = 1.$ This means that any two terms of this progression are coprime. \begin{exa} Prove that any two consecutive Fibonacci numbers are relatively prime.\end{exa} Solution: Let $d = (f_n, f_{n + 1}).$ As $f_{n + 1} - f_n = f_{n - 1}$ and $d$ divides the sinistral side of this equality, $d|f_{n - 1}.$ Thus $d|(f_n - f_{n - 1}) = f_{n - 2}$. Iterating on this process we deduce that $d|f_1 = 1$ and so $d = 1$. \\ {\em Aliter:} By Cassini's Identity $f_{n - 1}f_{n + 1} - f_n ^2 = (-1)^n.$ Thus $d|(-1)^n$, i.e., $d = 1.$ \begin{exa} Prove that $$ (f_m, f_n) = f_{(n, m)}.$$\end{exa} Solution: Set $d = (f_n, f_m), c = f_{(m, n)}, a = (m, n)$. We will prove that $c|d$ and $d|c.$ Since $a|m$ and $a|n$, $f_a|f_m$ and $f_a|f_n$ by Theorem \ref{thm:dividing_fibonacci}. Thus $$f_a|(f_m, f_m),$$ i.e., $c|d.$ \bigskip Now, by the Bachet-Bezout Theorem, there are integers $x, y$ such that $xm + yn = a.$ Observe that $x, y$ cannot be both negative, otherwise $a$ would be negative. As $a|n, a|m$ we have $a \leq n, a \leq m.$ They cannot be both positive since then $a = xm + yn \geq m + n,$ a contradiction. Thus they are of opposite signs, and we assume without loss of generality that $x \leq 0, y > 0.$ \bigskip Observe that $$ f_{yn} = f_{a - xm} = f_{a - 1}f_{-xm} + f_af_{-xm + 1}$$ upon using the identity $$ f_{s + t} = f_{s - 1}f_t + f_sf_{t + 1}$$of Theorem \ref{thm:cassini1}. As $n|yn, m|(-xm),$ we have that $f_n|f_{yn}, f_m|f_{-xm}$. This implies that $ (f_n, f_m)|f_{yn}$ and $ (f_n, f_m)|f_{-xm}.$ Hence $$ (f_n, f_m)|f_af_{-xm + 1}.$$We saw earlier that $(f_n, f_m)|f_{-xm}$. If it were the case that $$(f_n, f_m)|f_{-xm + 1},$$ then $(f_n, f_m)$ would be dividing two consecutive Fibonacci numbers, a contradiction to the preceding problem in the case when $(f_n, f_m) > 1.$ The case $= 1$ is a triviality. Therefore $(f_n, f_m)|f_a,$ which is what we wanted to prove. \begin{exa} Prove that no odd Fibonacci number is ever divisible by $17$.\end{exa} Solution: Let $d = (17, f_n)$, which obviously must be odd. Then $(17, f_n) = (34, f_n) = (f_9, f_n) = f_{(9, n)} = f_1, f_3$ or $f_9.$ This means that $d = (17, f_n) = 1, 2$ or 34. This forces $d = 1.$ \begin{exa} The {\em Catalan number of order} $n$ is defined as $$ C_n = \dfrac{1}{n + 1}\binom{2n}{n}.$$ Prove that $C_n$ is an integer for all natural numbers $n$. \end{exa} Solution: By the binomial absorption identity,$$ \dfrac{2n + 1}{n + 1}\binom{2n}{n} = \binom{2n + 1}{n + 1}.$$ Since $2n + 1$ and $n + 1$ are relatively prime, and since the dextral side is an integer, it must be the case that $n + 1$ divides $\displaystyle{\binom{2n}{n}}$. \\ \begin{exa} Let $n$ be a natural number. Find the greatest common divisor of $$ \binom{2n}{1}, \binom{2n}{3}, \ldots , \binom{2n}{2n - 1}.$$ \end{exa} Solution: Since $$ \sum _{k = 1} ^n \binom{2n}{2k - 1} = 2^{2n - 1},$$the gcd must be of the form $2^a$. Since the gcd must divide $\binom{2n}{1} = 2n$, we see that it has divide $2^{l + 1}$, where $l$ is the largest power of 2 that divides $n$. We claim that $2^{l + 1}$ divides all of them. We may write $n = 2^l m$, where ${\mathscr M}$ is odd. Now, $$ \binom{2^{l + 1}m}{2k - 1} = \dfrac{2^{l + 1}m}{2k - 1}\binom{2^{l + 1}m - 1}{2k - 2}.$$ But $2k - 1\not |2^{l + 1}$ for $k > 1$. This establishes the claim. \begin{exa} Let any fifty one integers be taken from amongst the numbers $1, 2, \ldots , 100.$ Show that there are two that are relatively prime. \end{exa} Solution: Arrange the 100 integers into the 50 sets $$ \{ 1, 2\} , \{ 3, 4 \} , \{ 5, 6 \} \ldots , \{ 99, 100 \} .$$ Since we are choosing fifty one integers, there must be two that will lie in the same set. Those two are relatively prime, as consecutive integers are relatively prime. \begin{exa} Prove that any natural number $n > 6$ can be written as the sum of two integers greater than $1$, each of the summands being relatively prime. \end{exa} Solution: If $n$ is odd, we may choose $a = 2, b = n - 2$. If $n$ is even, then is either of the form $4k$ or $4k + 2$. If $n = 4k,$ then take $a = 2k + 1, b = 2k - 1$. These two are clearly relatively prime (why?). If $n = 4k + 2, k > 1$ take $a = 2k + 3, b = 2k - 1.$ \\ \begin{exa} How many positive integers $\leq 1260$ are relatively prime to $1260$? \end{exa} Solution: As $1260 = 2^2 \cdot 3^2 \cdot 5\cdot 7,$ the problem amounts to finding those numbers less than 1260 which are not divisible by 2, 3, 5, or 7. Let $A$ denote the set of integers $\leq 1260$ which are multiples of 2, $B$ the set of multiples of 3, etc. By the Inclusion-Exclusion Principle, $$ \begin{array}{lcl}|A\cup B \cup C\cup D| & = & |A| + |B| + |C| + |D| \\ & & \quad - |A\cap B| - |A\cap C| - |A\cap D| \\ & & \quad - |B\cap C| - |B\cap D| - |C\cap D| \\ & & \quad + |A\cap B\cap C| + |A\cap B\cap D| + |A\cap C \cap D| \\ & & \quad + |B\cap C \cap D| - |A\cap B\cap C\cap D| \\ & = & 630 + 420 + 252 + 180 - 210 - 126 - 90 - 84 \\ & & \quad - 60 - 36 + 42 + 30 + 18 + 12 - 6 = 972.\end{array}$$ The number of integers sought is then $1260 - 972 = 288$. \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro} Show that $$ (a, b)[a, b] = ab$$ for all natural numbers $a, b$.\end{pro} \begin{pro} Find ${\rm lcm} \ (23!41!, 29!37!).$\end{pro} \begin{pro} Find two positive integers $a, b$ such that $$a^2 + b^2 = 85113, \ {\rm and \ lcm} \ (a, b) = 1764. $$\end{pro} \begin{pro}Find $a, b\in \BBN$ with $(a, b) = 12, [a, b] = 432.$\end{pro} \begin{pro} Prove that $(a, b)^n = (a^n, b^n)$ for all natural numbers $n$.\end{pro} \begin{pro}Let $a\in \BBN$. Find, with proof, all $b\in \BBN$ such that $$(2^b - 1)|(2^a + 1).$$\end{pro} \begin{pro} Show that $(n^3 + 3n + 1, 7n^3 + 18n^2 - n - 2) = 1.$\end{pro} \begin{pro} Let the integers $a_n, b_n$ be defined by the relation $$a_n + b_n\sqrt{2} = (1 + \sqrt{2})^n, \ n \in \BBN.$$Prove that $\gcd (a_n, b_n) = 1\ \forall \ n.$\end{pro} \begin{pro}Prove or disprove the following two propositions: \begin{enumerate} \item If $a, b \in \BBN , a < b,$ then in any set of $b$ consecutive integers there are two whose product is divisible by $ab$. \item If $a, b, c, \in \BBN , a < b < c,$ then in any set of $c$ consecutive integers there are three whose product is divisible by $abc.$ \end{enumerate}\end{pro} \begin{pro} Let $n, k, n \geq k > 0$ be integers. Prove that the greatest common divisor of the numbers $$ \binom{n}{k}, \binom{n + 1}{k}, \ldots , \binom{n + k}{k}$$is $1$. \end{pro} (Hint: Prove $$ \sum _{j = 0} ^k (-1)^j \binom{k}{j}\binom{n + j}{k} = (-1)^k .)$$ \begin{pro}Let $F_n = 2^{2^n} + 1$ be the $n$-th {\em Fermat number.} Find $(F_n , F_m)$.\end{pro} \begin{pro} Find the greatest common divisor of the sequence $$ 16^n + 10n - 1, \,\, n = 1, 2, \ldots .$$\end{pro} \begin{pro}Demonstrate that $(n! + 1, (n + 1)! + 1) = 1.$\end{pro} \begin{pro} Prove that any natural number $n > 17$ can be written as $n = a + b + c$ where $a, b, c $ are pairwise relatively prime natural numbers each exceeding $1$. \end{pro} (Hint: Consider $n\mod 12$. Write two of the summands in the form $6k + s$ and the third summand as a constant.) \begin{pro}Prove that there are no positive integers $a, b, n > 1$ with $$ (a^n - b^n )|(a^n + b^n ).$$\end{pro} \begin{pro} Prove that the binomial coefficients have the following hexagonal property: $$ \gcd\left(\binom{n - 1}{k - 1}, \binom{n}{k + 1}, \binom{n + 1}{k}\right)$$ equals $$ \gcd\left(\binom{n - 1}{k}, \binom{n + 1}{k + 1}, \binom{n}{k - 1}\right) .$$ \end{pro} \begin{pro}[Putnam, 1974] Call a set of integers {\em conspiratorial} if no three of them are pairwise relatively prime. What is the largest number of elements in any conspiratorial subset of the integers $1$ through $16$? \end{pro} \end{multicols} \section{Primes} Recall that a {\em prime number} is a positive integer greater than 1 whose only positive divisors are itself and 1. Clearly 2 is the only even prime and so 2 and 3 are the only consecutive integers which are prime. An integer different from 1 which is not prime is called {\em composite.} It is clear that if $n > 1$ is composite then we can write $n$ as $n = ab, 1 < a \leq b < n, a, b \in \BBN$. \begin{thm} If $n > 1$, then $n$ is divisible by at least one prime. \end{thm} \begin{pf} Since $n > 1,$ it has at least one divisor $> 1.$ By the Well Ordering Principle, $n$ must have a least positive divisor greater than 1, say $q$. We claim that $q$ is prime. For if not then we can write $q$ as $q = ab, 1 < a \leq b < q$. But then $a$ is a divisor of $n$ greater than 1 and smaller than $q$, which contradicts the minimality of $q$.\end{pf} \begin{thm}[Euclid]\label{thm:infinitude_primes} There are infinitely many primes. \end{thm} \begin{pf} Let $p_1, p_2, \ldots p_k$ be a list of primes. Construct the integer $$ n = p_1 p_2 \cdots p_k + 1.$$This integer is greater than $1$ and so by the preceding problem, it must have a prime divisor $p.$ Observe that $p$ must be different from any of $p_1, p_2, \ldots , p_k$ since $n$ leaves remainder $1$ upon division by any of the $p_i$. Thus we have shown that no finite list of primes exhausts the set of primes, i.e., that the set of primes is infinite.\end{pf} \begin{lem} The product of two numbers of the form $4k + 1$ is again of that form.\end{lem} \begin{pf} $(4a + 1)(4b + 1) = 4(4ab + a + b) + 1$.\end{pf} \begin{thm}\label{thm:infinitude_primes_4n+3} There are infinitely many primes of the form $4n + 3.$\end{thm} \begin{pf} Any prime either equals $2$, or is of the form $4k \pm 1.$ We will show that the collection of primes of the form $4k - 1$ is inexhaustible. Let $$\{ p_1 , p_2 , \ldots p_n\}$$ be any finite collection of primes of the form $4k - 1.$ Construct the number $$ N = 4p_1 p_2 \cdots p_n - 1.$$Since each $p_k$ is $ \geq 3, N \geq 11.$ Observe that $N$ is not divisible by any of the primes in our collection. Now either $N$ is a prime, in which case it is a prime of the form $4k - 1$ not on the list, or it is a product of primes. In the latter case, all of the prime factors of $N$ cannot be of the form $4k + 1$, for the product of any two primes of this form is again of this form, in view of the preceding problem. Thus $N$ must be divisible by some prime of the form $4k - 1$ not on the list. We have thus shown that given any finite list of primes of the form $4k - 1$ we can always construct an integer which is divisible by some prime of the form $4k -1$ not on that list. The assertion follows. \end{pf} \begin{exa} Prove that there are arbitrarily long strings that do not contain a prime number. \end{exa} Solution: Let $k \in \BBN , k \geq 2$. Then each of the numbers $$ k! + 2, \ldots , k! + k$$is composite. \begin{thm} If the positive integer $n$ is composite, then it must have a prime factor $p$ with $p \leq \sqrt{n}$. \end{thm} \begin{pf} Suppose that $n = ab, 1 < a \leq b < n.$ If both $a$ and $b$ are $> \sqrt{n}$, then $n = ab > \sqrt{n}\sqrt{n} = n,$ a contradiction. Thus $n$ has a factor $\neq 1$ and $\leq \sqrt{n}$, and hence a prime factor, which is $\leq \sqrt{n}.$ \end{pf} \begin{exa} Find the number of prime numbers $\leq 100.$ \end{exa} Solution: Observe that $\sqrt{100} = 10.$ By the preceding theorem, all the composite numbers in the range $10 \leq n \leq 100$ have a prime factor amongst $2, 3, 5,$ or 7. Let $A_m$ denote the multiples of ${\mathscr M}$ which are $\leq 100$. Then $|A_2| = 50, |A_3| = 33, |A_5| = 20, |A_7| = 14, |A_6| = 16, |A_{10}| = 10, |A_{14}| = 7, |A_{15}| = 6, |A_{21}| = 4, |A_{35}| = 2, |A_{30}| = 3, |A_{42}| = 2, |A_{70}| = 1, |A_{105}| = 0, |A_{210}| = 0.$ Thus the number of primes $\leq 100$ is $$\begin{array}{ll} = & 100 - ({\rm \ number\ of\ composites\ \leq 1}) - 1 \\ = & 4 + 100 - {\rm \ multiples\ of\ 2, \ 3, \ 5, or \ 7} \leq 100 - 1 \\ = & 4 + 100 - (50 + 33 + 20 + 14) + (16 + 10 + 7 + 6 + 4 + 2) \\ & \quad - (3 + 2 + 1 + 0) - 0 - 1 \\ = & 25, \end{array}$$where we have subtracted the 1, because 1 is neither prime nor composite. \begin{lem} If $p$ is a prime, $\displaystyle{\binom{p}{k}}$ is divisible by $p$ for all $0 < k < p.$ \end{lem} \begin{pf} $$ \binom{p}{k} = \dfrac{p(p - 1)\cdots (p - k + 1)}{k!}$$yields $$ k!\binom{p}{k} = p(p - 1)\cdots (p - k + 1),$$whence $p|k!\displaystyle{\binom{p}{k}}$. Now, as $k < p, p\not |k!$. By Euclid's Lemma, it must be the case that $p|\displaystyle{\binom{p}{k}}$.\end{pf} \begin{exa} Prove that if $p$ is a prime, then $p$ divides $2^p - 2.$\end{exa} Solution: By the Binomial Theorem: $$ 2^p - 2 = (1 + 1)^p - 2 = \binom{p}{1} + \binom{p}{2} + \cdots + \binom{p}{p - 1},$$as $\binom{p}{0} = \binom{p}{p} = 1.$ By the preceding lemma, $p$ divides each of the terms on the dextral side of the above. This establishes the assertion. \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro} Prove that there are infinitely many primes of the form $6n + 5.$\end{pro} \begin{pro} Use the preceding problem to show that there are infinitely many primes $p$ such that $p - 2$ is not a prime. \end{pro} \begin{pro}If $p$ and $q$ are consecutive odd primes, prove that the prime factorisation of $p + q$ has at least three (not necessarily distinct) primes.\end{pro} \begin{pro} \begin{enumerate} \item Let $p$ be a prime and let $n\in \BBN$. Prove, by induction on $n$, that $p|(n^p - n)$. \item Extend this result to all $n\in \BBZ$. \item Prove {\em Fermat's Little Theorem}: if $p\not\! |n$, then $p|(n^{p - 1} - 1).$ \item Prove that $42|n^7 - n, n \in \BBZ$. \item Prove that $30|n^5 - n, n \in \BBZ$.\end{enumerate}\end{pro} \begin{pro} Let $p$ be an odd prime and let $(a, b) = 1$. Prove that $$ \left( a + b, \dfrac{a^p + b^p}{a + b}\right) \ {\rm divides} \ p.$$\end{pro} \begin{pro}Prove that $3, 5, 7$ is the only prime triplet of the form $p, p + 2, p + 4$. \end{pro} \begin{pro} Let $n > 2$. Prove that if one of the numbers $2^n - 1$ and $2^n + 1$ is prime, then the other is composite.\end{pro} \end{multicols} \section{Fundamental Theorem of Arithmetic} Consider the integer $1332.$ It is clearly divisible by $2$ and so we obtain $1332 = 2\cdot 666.$ Now, $666$ is clearly divisible by $6$, and so $1332 = 2\cdot 2\cdot 3\cdot 111.$ Finally, $111$ is also divisible by $3$ and so we obtain $1332 = 2\cdot 2\cdot 3\cdot 3\cdot 37.$ We cannot further decompose $1332$ as a product of positive integers greater than $1$, as all $2, 3, 37$ are prime. We will show now that such decomposition is always possible for a positive integer greater than $1$. \begin{thm} Every integer greater than $1$ is a product of prime numbers.\end{thm} \begin{pf} Let $n > 1.$ If $n$ is a prime, then we have nothing to prove. Assume that $n$ is composite and let $q_1$ be its least proper divisor. By Theorem 4.5, $q_1$ is a prime. Set $n = q_1n_1, 1 < n_1 < n.$ If $n_1$ is a prime, then we arrived at the result. Otherwise, assume that $n_1$ is composite, and let $q_2$ be its least prime divisor, as guaranteed by Theorem 4.5. We can write then $n = q_1q_2n_2, 1 < n_2 < n_1 < n.$ Continuing the argument, we arrive at a chain $n > n_1 > n_2 \cdots > 1$, and this process must stop before $n$ steps, as $n$ is a positive integer. Eventually we then have $n = q_1q_2 \cdots q_s$. \end{pf} We may arrange the prime factorisation obtained in the preceding Theorem as follows, $$ n = p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}, \ \ a_1 > 0, a_2 > 0, \ldots , a_k > 0, $$ $$ p_1 < p_2 < \cdots < p_k, $$where the $p_j$ are primes. We call the preceding factorisation of $n$, the {\em canonical factorisation} of $n$. For example $2^33^25^27^3$ is the canonical factorisation of $617400$. \begin{thm}[Fundamental Theorem of Arithmetic]\label{thm:fundamental_arithmetic} Every integer $> 1$ can be represented as a product of primes in only one way, apart from the order of the factors. \end{thm} \begin{pf} We prove that a positive integer greater than 1 can only have one canonical factorisation. Assume that $$ n = p_1^{a_1}p_2^{a_2} \cdots p_s^{a_s} = q_1^{b_1}q_2^{b_2} \cdots q_t^{b_t}$$are two canonical factorisations of $n.$ By Euclid's Lemma (example 1.2) we conclude that every $p$ must be a $q$ and every $q$ must be a $p$. This implies that $s = t$. Also, from $p_1 < p_2 < \cdots < p_s$ and $q_1 < q_2 < \cdots < q_t$ we conclude that $p_j = q_j, 1 \leq j \leq s.$ If $a_j > b_j$ for some $j$ then, upon dividing by $p_j^{b_j}$, we obtain $$p_1^{a_1}p_2^{a_2}\cdots p_j^{a_j - b_j}\cdots p_s^{a_s} = p_1^{b_1}p_2^{b_2}\cdots p_{j - 1}^{b_{j - 1}}p_{j + 1}^{b_{j + 1}} \cdots p_s^{b_s},$$which is impossible, as the sinistral side is divisible by $p_j$ and the dextral side is not. Similarly, the alternative $a_j < b_j$ for some $j$ is ruled out and so $a_j = b_j$ for all $j$. This finishes the proof. \end{pf} It is easily seen, by the Fundamental Theorem of Arithmetic, that if $a$ has the prime factorisation $a = p_1 ^{a_1}p_2 ^{a_2} \cdots p_n ^{a_n}$ and $b$ has the prime factorisation $b = p_1 ^{b_1}p_2 ^{b_2}\cdots p_n ^{b_n}$, (it may be the case that some of the $a_k$ and some of the $b_k$ are zero) then \begin{equation}(a, b) = p_1 ^{\min (a_1 , b_1)} p_2 ^{\min (a_2 , b_2 )} \cdots p_n ^{\min (a_n , b_n )}.\end{equation} and also \begin{equation}[a, b] = p_1 ^{\max (a_1 , b_1)} p_2 ^{\max (a_2 , b_2 )} \cdots p_n ^{\max (a_n , b_n )}.\end{equation} Since $x + y = \max (x, y) + \min (x, y),$ it clearly follows that $$ ab = (a, b)[a, b]. $$ \begin{exa} Prove that $\sqrt{2}$ is irrational. \end{exa} Solution: Assume that $\sqrt{2} = a/b$ with relatively prime natural numbers $a, b$. Then $2b^2 = a^2$. The sinistral side of this last equality has an odd number of prime factors (including repetitions), whereas the dextral side has an even number of prime factors. This contradicts the Fundamental Theorem of Arithmetic. \begin{exa} Prove that if the polynomial $$ p(x) = a_0 x^n + a_1 x^{n - 1} + \cdots + a_{n - 1}x + a_n $$ with integral coefficients assumes the value $7$ for four integral values of $x$, then it cannot take the value $14$ for any integral value of $x$. \end{exa} Solution: First observe that the integer 7 can be decomposed into at most three different integer factors $7 = -7(1)(-1)$. Assume that $p(a_k ) - 7 = 0$ for distinct $a_k , 1 \leq k \leq 4.$ Then $$ p(x) - 7 = (x - a_1 )(x - a_2 )(x - a_3 )(x - a_4 )q(x)$$ for a polynomial $q$ with integer coefficients. Assume that there is an integer ${\mathscr M}$ with $p(m) = 14.$ Then $$ 7 = p(m) - 7 = (m - a_1 )(m - a_2 )(m - a_3 )(m - a_4 )q(m).$$ Since the factors $m - a_k$ are all distinct, we have decomposed the integer 7 into at least four different factors. This is impossible, by the Fundamental Theorem of Arithmetic. \begin{exa} Prove that the product of three consecutive integers is never a perfect power (i.e., a perfect square or a perfect cube, etc.). \end{exa} Solution: Let the integer be $(n - 1)n(n + 1) = (n^2 - 1)n$. Since $n^2 - 1$ and $n$ are relatively prime, by the Fundamental Theorem of Arithmetic, $n^2 - 1$ is a perfect $k$th power $(k \geq 2)$ and $n$ is also a perfect $k$th power. But then, $n^2 - 1$ and $n^2$ would be {\em consecutive} perfect $k$th powers, sheer nonsense. \begin{exa} Prove that $m^5 + 3m^4 n - 5m^3 n^2 - 15m^2 n^3 + 4mn^4 + 12n^5$ is never equal to $33$. \end{exa} Solution: Observe that $$ m^5 + 3m^4 n - 5m^3 n^2 - 15m^2 n^3 + 4mn^4 + 12n^5 $$ $$ \qquad = (m - 2n)(m - n)(m + n)(m + 2n)(m + 3n).$$ Now, 33 can be decomposed as the product of at most four different integers $33 = (-11)(3)(1)(-1)$. If $n \neq 0$, the factors in the above product are all different. They cannot be multiply to 33, by the Fundamental Theorem of Arithmetic, as 33 is the product of $4$ different factors and the expression above is the product of $5$ different factors for $n \neq 0.$. If $n = 0,$ the product of the factors is $m^5$, and 33 is clearly not a fifth power. \begin{exa} Prove that the sum $$ S = 1/2 + 1/3 + 1/4 + \cdots + 1/n$$ is never an integer. \end{exa} Solution: Let $k$ be the largest integer such that $2^k \leq n,$ and $P$ the product of all the odd natural numbers not exceeding $n$. The number $2^{k - 1}PS$ is a sum, all whose terms, except for $2^{k - 1}P\dfrac{1}{2^k},$ are integers. \begin{exa}Prove that there is exactly one natural number n for with $2^8 + 2^{11} + 2^n$ is a perfect square.\end{exa} Solution: If $k^2 = 2^8 + 2^{11} + 2^n = 2304 + 2^n = 48^2 + 2^n,$ then $k^2 - 48^2 = (k - 48)(k + 48) = 2^n$. By unique factorisation, $k - 48 = 2^s, k + 48 = 2^t, s + t = n.$ But then $2^t - 2^s = 96 = 3\cdot 2^5$ or $2^s(2^{t - s} - 1) = 3 \cdot 2^5.$ By unique factorisation, $s = 5, t - s = 2,$ giving $s + t = n = 12.$ \begin{exa} Prove that in any set of $33$ distinct integers with prime factors amongst $\{ 5, 7, 11, 13, 23\}$, there must be two whose product is a square. \end{exa} Solution: Any number in our set is going to have the form $$ 5^a 7^b 11^c 13^d 23^f .$$ Thus to each number in the set, we associate a vector $(a, b, c, d, f)$. These vectors come in 32 different flavours, according to the parity of the components. For example (even, odd, odd, even, odd) is one such class. Since we have 33 integers, two (at least) will have the same parity in their exponents, and the product of these two will be a square. \begin{exa}[IMO, 1985] Given a set ${\mathscr M}$ of $1985$ distinct positive integers, none with a prime factor greater than $26$, prove that ${\mathscr M}$ contains a subset of four distinct elements whose product is the fourth power of an integer. \end{exa} Solution: Any number in our set is going to be of the form $$ 2^a 3^b 5^c7^d11^f13^g17^h19^j23^k.$$Thus if we gather 513 of these numbers, we will have two different ones whose product is a square. Start weeding out squares. Since we have $1985 > 513$ numbers, we can find a pair of distinct $a_1, b_1$ such that $a_1b_1 = c_1 ^2.$ Delete this pair. From the $1983$ integers remaining, we can find a pair of distinct $a_2, b_2$ such that $a_2b_2 = c_2 ^2.$ Delete this pair. From the $1981$ integers remaining, we can find a pair $a_3, b_3$ such that $a_3b_3 = c_3 ^2.$ We can continue this operation as long as we have at least $513$ integers. Thus we may perform this operation $n + 1$ times, were $n$ is the largest positive integer such that $1985 - 2n \geq 513,$ i.e., $n = 736$. Therefore, we are able to gather $737$ pairs $a_k, b_k$ such that $a_kb_k = c_k ^2$. Now, the $737$ numbers $c_k$ have all their prime factors smaller than $26$, and since $737 > 513,$ we may find two distinct $c_m$ say $c_i$ and $c_j, i \neq j$, such that $c_ic_j = a^2,$ a perfect square. But then $c_ic_j = a^2$ implies that $a_ib_ia_jb_j = a^4,$ a fourth power. Thus we have found four distinct numbers in our set whose product is a fourth power. \begin{exa} Let any fifty one integers be taken from amongst the numbers $1, 2, \ldots , 100.$ Show that there must be one that divides some other. \end{exa} Solution: Any of the fifty one integers can be written in the form $2^a m$, where $m$ is odd. Since there are only fifty odd integers between 1 and 100, there are only fifty possibilities for $m$. Thus two (at least) of the integers chosen must share the same odd part, and thus the smaller will divide the larger. \begin{exa}[USAMO 1972] Prove that $$ \dfrac{[a, b, c]^2}{[a, b][b, c][c, a]} = \dfrac{(a, b, c)^2}{(a, b)(b, c)(c, a)}.$$ \end{exa} Solution: Put $$ a = \prod p_k ^{\alpha _k}, \ b = \prod p_k ^{\beta _k}, \ c = \prod p_k ^{\gamma _k},$$with primes $p_k.$ The assertion is equivalent to showing $$ 2\max ( \alpha _k , \beta _k , \gamma _k ) - \max (\alpha _k , \beta _k) - \max (\alpha _k , \gamma _k ) - \max ( \beta _k , \gamma _k )$$ $$ \qquad = 2\min ( \alpha _k , \beta _k , \gamma _k ) - \min (\alpha _k , \beta _k) - \min (\alpha _k , \gamma _k ) - \min ( \beta _k , \gamma _k ).$$By symmetry, we may assume, without loss of generality, that $\alpha _k \geq \beta _k \geq \gamma _k$. The equation to be established reduces thus to the identity $$ 2\alpha _k - \alpha _k - \alpha _k - \beta _k = 2\gamma _k - \beta _k - \gamma _k - \gamma _k .$$ \begin{exa}Prove that $n = 24$ is the largest natural number divisible by all integral $a, 1 \leq a \leq \sqrt{n}.$\end{exa} Solution: Suppose $n$ is divisible by all the integers $\leq \sqrt{n}$. Let $p_1 = 2, p_2 = 3, \ldots , p_l$ be all the primes $\leq \sqrt{n}$, and let $k_j$ be the unique integers such that $p_j ^{k_j} \leq \sqrt{n} < p_j ^{k_j + 1}$. Clearly $n^{l/2} < p_1 ^{k_1 + 1} p_2 ^{k_2+ 1}\cdots p_l ^{k_l + 1}$. Let lcm$(1, 2, 3, \ldots , \floor{ \sqrt{n}} - 1, \floor{ \sqrt{n} }) = K$. Clearly then $K = p_1 ^{k_1} p_2 ^{k_2} \cdots p_l ^{k_l}.$ Hence $p_1 ^{k_1 + 1}p_2 ^{k_2 + 1} \cdots p_l ^{k_l + 1} \leq K^2$ and thus $n^{l/2} < K^2.$ By hypothesis, $n$ must be divisible by $K$ and so $K \leq n.$ Consequently, $n^{l/2} < n^2.$ This implies that $l < 4$ and so $n < 49.$ By inspection, we see that the only valid values for $n$ are $n = 2, 4, 6, 8, 12, 24.$ \begin{exa}[Irving Kaplansky] A positive integer $n$ has the property that for $0 < l < m < n,$ $$ S = l + (l + 1) + \ldots + m$$is never divisible by $n$. Prove that this is possible if and only if $n$ is a power of $2$.\end{exa} Solution: Set $n = s2^k$ with $s$ odd. If $s = 1, 2S = (l + m)(m - l + 1)$, which has one factor even and one factor odd, cannot be divisible by $2n = 2^{k + 1}$, since, its even factor is less than $2n$. But if $s > 1,$ then $S$ is divisible by $n$, with $0 < l < m < n,$ if we take $$ m = (s + 2^{k + 1} - 1)/2$$and $$ l = \left\{\begin{array}{ll} 1 + m - 2^{k + 1}, & s > 2^{k + 1}, \\ 1 + m - s, & s < 2^{k + 1}. \end{array} \right.$$ \begin{exa} Let $0 < a_1 < a_2 < \cdots < a_k \leq n,$ where $k > \floor{ \dfrac{n + 1}{2} },$ be integers. Prove that $$ a_1 + a_j = a_r$$is soluble.\end{exa} Solution: The $k - 1$ positive integers $a_i - a_1 , 2 \leq i \leq k$, are clearly distinct. These, together with the $k$ given distinct $a$'s, give $2k - 1 > n$ positive integers, each not greater than $n$. Hence, at least one of the integers is common to both sets, so that at least once $a_r - a_1 = a_j$. The sequence $\floor{ n/2 } + 1, \floor{ n/2 } + 2, \ldots , n,$ shows that for $k = \floor{ (n + 1)/2 } $ the result is false. \begin{exa} Let $0< a_1 < a_2 < \cdots < a_n \leq 2n$ be integers such that the least common multiple of any two exceeds $2n$. Prove that $a_1 > \floor{ \dfrac{2n}{3} }.$\end{exa} Solution: It is clear that no one of the numbers can divide another (otherwise we would have an lcm $\leq 2n$). Hence, writing $a_k = 2^{t_k}A_k , \ A_k$ odd, we see that all the $A_k$ are different. Since there are $n$ of them, they coincide in some order with the set of all positive odd numbers less than $2n$. Now, consider $a_1 = 2^{t_1}A_1$. If $a_1 \leq \floor{ 2n/3 } $, then $3a_1 = 2^{t_1}3A_1 \leq 2n,$ and $3A_1 < 2n$. Since $3A_1$ would then be an odd number $< 2n$, $3A_1 = A_j$ for some $j$, and $a_j = 2^{t_j}3A_1$. Thus either $[ a_1 , a_j] = 2^{t_1}3A_1 = 3a_1 \leq 2n,$ or $[ a_1 , a_j] = 2^{t_j}3A_1 = a_j \leq 2n$. These contradictions establish the assertion. \begin{exa}[Putnam, 1980] Derive a formula for the number of quadruples $(a, b, c, d)$ such that $$ 3^r 7^s = [a, b, c] = [b, c, d] = [c, d, a] = [d, a, b].$$ \end{exa} Solution: By unique factorisation, each of $a, b, c, d$ must be of the form $3^m 7^n, 0 \leq m \leq r, 0 \leq n \leq s$. Moreover, ${\mathscr M}$ must equal $r$ for at least two of the four numbers, and $n$ must equal $s$ for at least two of the four numbers. There are $\binom{4}{2}r^2 = 6r^2$ ways of choosing exactly two of the four numbers to have exponent $r$, $\binom{4}{3}r = 4r$ ways of choosing exactly three to have exponent $r$ and $\binom{4}{4} = 1$ of choosing the four to have exponent $r$. Thus there is a total of $1 + 4r + 6r^2$ of choosing at least two of the four numbers to have exponent $r$. Similarly, there are $1 + 4s + 6s^2$ ways of choosing at least two of the four numbers to have exponent $s$. The required formula is thus $$ (1 + 4r + 6r^2 )(1 + 4s + 6s^2).$$ \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro} Prove that $\log _{10} 7$ is irrational. \end{pro} \begin{pro} Prove that $$ \dfrac{\log 3}{\log 2}$$is irrational.\end{pro} \begin{pro} Find the smallest positive integer such that $n/2$ is a square and $n/3$ is a cube. \end{pro} \begin{pro} How many integers from $1$ to $10^{20}$ inclusive, are not perfect squares, perfect cubes, or perfect fifth powers?\end{pro} \begin{pro} Prove that the sum $$ 1/3 + 1/5 + 1/7 + \cdots + 1/(2n + 1)$$ is never an integer. \end{pro} (Hint: Look at the largest power of 3 $\leq n$). \begin{pro} Find $\min _{k \geq 1} 36^k - 5^k$.\end{pro} (Hint: Why is $36^k - 1 - 5^k \neq 0$?) \begin{pro}[AIME 1987] Find the number of ordered triples $(a, b, c)$ of positive integers for which $[a, b] = 1000, [b, c] = [a, c] = 2000.$\end{pro} \begin{pro} Find the number of ways of factoring $1332$ as the product of two positive relatively prime factors each greater than $1$. Factorisations differing in order are considered the same.\end{pro} Answer: 3. \begin{pro} Let $p_1, p_2, \ldots , p_t$ be different primes and $a_1, a_2, \ldots a_t$ be natural numbers. Find the number of ways of factoring $p_1^{a_1}p_2^{a_2}\cdots p_t^{a_t}$ as the product of two positive relatively prime factors each greater than $1$. Factorisations differing in order are considered the same.\end{pro} Answer: $2^{t - 1} - 1$. \begin{pro} Let $n = p_1^{a_1}p_2^{a_2}\cdots p_t^{a_t}$ and $ m = p_1^{b_1}p_2^{b_2}\cdots p_t^{b_t}$, the $p$'s being different primes. Find the number of the common factors of $m$ and $n.$\end{pro} Answer: $$ \prod _{k = 1} ^t (1 + \min (a_k, b_k)).$$ \begin{pro}[USAMO 1973] Show that the cube roots of three distinct prime numbers cannot be three terms (not necessarily consecutive) of an arithmetic progression. \end{pro} \begin{pro}Let $2 = p_1 , 3 = p_2 , \ldots $ be the primes in their natural order and suppose that $n \geq 10$ and that $1 < j < n$. Set $$ N_1 = p_1 p_2 \cdots p_{j - 1} - 1, N_2 = 2p_1 p_2 \cdots p_{j - 1} - 1, \ldots$$ and $$ N_{p_j} = p_j p_1 p_2 \cdots p_{j - 1} - 1$$Prove\begin{enumerate} \item Each $p_i , j \leq i \leq n$, divides at most one of the $N_{p_k}, 1 \leq k \leq j $ \item There is a $j, 1 < j < n,$ for which $p_j > n - j + 1.$ \item Let $s$ be the smallest $j$ for which $p_j > n - j + 1.$ There is a $t, 1 \leq t \leq p_s$, such that all of $p_1 , \ldots p_n$ fail to divide $tp_1 p_2 \cdots p_{s - 1} - 1,$ and hence $p_{n + 1} < p_1 p_2 \cdots p_s$. \item The $s$ above is $> 4$ and so $p_{s - 1} - 2\geq s$ and $p_1 p_2 \cdots p_s < p_{s + 1} \cdots p_n$. \item (Bonse's Inequality) For $n \geq 4,$ $p_{n + 1} ^2 < p_1 \cdots p_{n}$. \end{enumerate}\end{pro} \begin{pro} Prove that $30$ is the only integer $n$ with the following property: if $1 \leq t \leq n$ and $(t, n) = 1$, then $t$ is prime.\end{pro} \begin{pro}[USAMO 1984]\begin{enumerate} \item For which positive integers $n$ is there a finite set $S_n$ of $n$ distinct positive integers such that the geometric mean of any subset of $S_n$ is an integer? \item Is there an {\em infinite} set $S$ of distinct positive integers such that the geometric mean of any finite subset of $S$ is an integer. \end{enumerate}\end{pro} \begin{pro}\begin{enumerate} \item {\bf (Putnam 1955)} Prove that there is no triplet of integers $(a, b, c)$, except for $(a, b, c) = (0, 0, 0)$ for which $$a + b\sqrt{2} + c\sqrt{3} = 0.$$ \item {\bf (Putnam 1980)} Prove that there exist integers a, b, c, not all zero and each of absolute value less than a million, such that $$ |a + b\sqrt{2} + c\sqrt{3}| < 10^{-11}.$$ \item {\bf (Putnam 1980)} Let $a, b, c$ be integers, not all zero and each of absolute value less than a million. Prove that $$ |a + b\sqrt{2} + c\sqrt{3}| > 10^{-21}.$$\end{enumerate}\end{pro} \begin{pro}[E\H{o}tv\H{o}s 1906] Let $a_1 , a_2 , \ldots , a_n$ be any permutation of the numbers $1, 2, \ldots , n$. Prove that if $n$ is odd, the product $$ (a_1 - 1)(a_2 - 2) \cdots (a_n - n)$$ is an even number. \end{pro} \begin{pro} Prove that from any sequence formed by arranging in a certain way the numbers from $1$ to $101$, it is always possible to choose $11$ numbers (which must not necessarily be consecutive members of the sequence) which form an increasing or a decreasing sequence.\end{pro} \begin{pro} Prove that from any fifty two integers it is always to choose two, whose sum, or else, whose difference, is divisible by $100.$\end{pro} \begin{pro}Prove that from any one hundred integers it is always possible to choose several numbers (or perhaps, one number) whose sum is divisible by $100.$\end{pro} \begin{pro} Given n numbers $x_1 , x_2 , \ldots , x_n$ each of which is equal to $\pm 1$, prove that if $$ x_1 x_2 + x_2 x_3 + \cdots + x_n x_1 = 0,$$ then $n$ is a multiple of $4$.\end{pro} \end{multicols} \chapter{Linear Diophantine Equations} \section{Euclidean Algorithm} We now examine a procedure that avoids factorising two integers in order to obtain their greatest common divisor. It is called the {\em Euclidean Algorithm} and it is described as follows. Let $a, b$ be positive integers. After using the Division Algorithm repeatedly, we find the sequence of equalities \begin{equation} \begin{array}{lcll} a & = & bq_1 + r_2, & 0 < r_2 < b, \\ b & = & r_2q_2 + r_3 & 0 < r_3 < r_2, \\ r_2 & = & r_3q_3 + r_4 & 0 < r_4 < r_3, \\ \vdots & \vdots & \vdots & \vdots \\ r_{n - 2} & = & r_{n - 1}q_{n - 1} + r_n & 0 < r_n < r_{n - 1}, \\ r_{n - 1} & = & r_nq_n. & \\ \end{array}\label{eq:euclidean_algorithm}\end{equation} The sequence of remainders will eventually reach a $r_{n + 1}$ which will be zero, since $b, r_2 , r_3, \ldots$ is a monotonically decreasing sequence of integers, and cannot contain more than $b$ positive terms. The Euclidean Algorithm rests on the fact, to be proved below, that $(a, b) = (b, r_2) = (r_2, r_3) = \cdots = (r_{n - 1}, r_n) = r_n.$ \begin{thm}\label{thm:gcd_ab_arith_prog}Prove that if $a, b, n$ are positive integers, then $$ (a, b) = (a + nb, b).$$ \end{thm} \begin{pf} Set $d = (a, b), c = (a + nb, b).$ As $d|a, d|b$, it follows that $d|(a + nb).$ Thus $d$ is a common divisor of both $(a + nb)$ and $b$. This implies that $d|c.$ On the other hand, $c|(a + nb), c|b$ imply that $c|((a + nb) - nb) = a.$ Thus $c$ is a common divisor of $a$ and $b$, implying that $c|d$. This completes the proof. \end{pf} \begin{exa} Use Theorem \ref{thm:gcd_ab_arith_prog} to find $(3456, 246)$.\end{exa} Solution: $(3456, 246) = (13\cdot 246 + 158, 246) = (158, 246),$ by the preceding example. Now, $(158, 246) = (158, 158 + 88) = (88, 158).$ Finally, $(88, 158) = (70, 88) = (18, 70) = (16, 18) = (2, 16) = 2$. Hence $(3456, 246) = 2.$ \begin{thm} If $r_n$ is the last non-zero remainder found in the process of the Euclidean Algorithm, then $$ r_n = (a, b).$$\end{thm} \begin{pf} From equations \ref{eq:euclidean_algorithm} $$ \begin{array}{lcl} r_2 & = & a - bq_1 \\ r_3 & = & b - r_2q_2 \\ r_4 & = & r_2 - r_3q_3 \\ \vdots & \vdots & \vdots \\ r_{n} & = & r_{n - 2} - r_{n - 1}q_{n - 1} \\ \end{array}$$ Let $r = (a, b)$. From the first equation, $r|r_2.$ From the second equation, $r|r_3.$ Upon iterating the process, we see that $r|r_n.$ But starting at the last equation \ref{eq:euclidean_algorithm} and working up, we see that $r_n|r_{n - 1}, r_n|r_{n - 2}, \ldots r_n|r_2, r_n|b, r_n|a$. Thus $r_n$ is a common divisor of $a$ and $b$ and so $r_n|(a, b).$ This gives the desired result. \end{pf} \begin{exa} Find $(23, 29)$ by means of the Euclidean Algorithm.\end{exa} Solution: We have $$ 29 = 1\cdot 23 + 6,$$ $$ 23 = 3\cdot 6 + 5,$$ $$ 6 = 1\cdot 5 + 1,$$ $$ 5 = 5\cdot 1. $$The last non-zero remainder is $1$, thus $(23, 29) = 1.$ An equation which requires integer solutions is called a {\em diophantine equation}. By the Bachet-Bezout Theorem, we see that the linear diophantine equation $$ ax + by = c$$has a solution in integers if and only if $(a, b)|c.$ The Euclidean Algorithm is an efficient means to find a solution to this equation. \begin{exa}\label{exa:lin_dio1} Find integers $x, y$ that satisfy the linear diophantine equation $$ 23x + 29y = 1.$$ \end{exa} Solution: We work upwards, starting from the penultimate equality in the preceding problem: $$ 1 = 6 - 1\cdot 5,$$ $$ 5 = 23 - 3 \cdot 6,$$ $$ 6 = 29\cdot 1 - 23.$$ Hence, $$ \begin{array}{lcl} 1 & = & 6 - 1\cdot 5 \\ & = & 6 - 1\cdot (23 - 3\cdot 6) \\ & = & 4\cdot 6 - 1\cdot 23 \\ & = & 4(29\cdot 1 - 23) - 1\cdot 23 \\ & = & 4\cdot 29 - 5\cdot 23. \end{array}$$This solves the equation, with $x = - 5, y = 4.$ \begin{exa} Find integer solutions to $$23x + 29y = 7.$$\end{exa} Solution: From the preceding example, $23(-5) + 29(4) = 1$. Multiplying both sides of this equality by $7$, $$ 23(-35) + 29(28) = 7,$$which solves the problem. \begin{exa} Find infinitely many integer solutions to $$ 23x + 29y = 1.$$ \end{exa} Solution: By Example \ref{exa:lin_dio1}, the pair $x_0 = -5, y_0 = 4$ is a solution. We can find a family of solutions by letting $$ x = -5 + 29t, \ y = 4 - 23t, \ \ t \in \BBZ.$$ \begin{exa} Can you find integers $x$, $y$ such that $3456x + 246y = 73$?\end{exa} Solution: No. $(3456, 246) = 2$ and $2\not |73.$ \begin{thm}\label{thm:solutions_linear_diophantine} Assume that $a$, $b$, $c$ are integers such that $(a, b)|c.$ Then given any solution $(x_0, y_0)$ of the linear diophantine equation $$ax + by = c$$any other solution of this equation will have the form $$ x = x_0 + t\dfrac{b}{d}, \ y = y_0 - t\dfrac{a}{d},$$where $d = (a, b)$ and $t \in \BBZ.$\end{thm} \begin{pf} It is clear that if $(x_0, y_0)$ is a solution of $ax + by = c,$ then $x = x_0 + tb/d, y = y_0 - ta/d$ is also a solution. Let us prove that any solution will have this form. Let $(x', y')$ satisfy $ax' + by' = c.$ As $ax_0 + by_0 = c$ also, we have $$a(x' - x_0) = b(y_0 - y').$$Dividing by $d = (a, b),$ $$ \dfrac{a}{d}(x' - x_0) = \dfrac{b}{d}(y_0 - y').$$ Since $(a/d, b/d) = 1,$ $\dfrac{a}{d}|(y_0 - y')$, in virtue of Euclid's Lemma. Thus there is an integer $t$ such that $t\dfrac{a}{d} = y_0 - y'$, that is, $y = y_0 - ta/d.$ From this $$ \dfrac{a}{d}(x' - x_0) = \dfrac{b}{d}t\dfrac{a}{d},$$which is to say $x' = x_0 + tb/d.$ This finishes the proof. \end{pf} \begin{exa} Find all solutions in integers to $$ 3456x + 246y = 234.$$\end{exa} Solution: By inspection, $3456(-1) + 246(15) = 234.$ By Theorem \ref{thm:solutions_linear_diophantine}, all the solutions are given by $x = -1 + 123t, y = 15 - 1728t, t \in \BBZ.$ \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro} Find the following: \begin{enumerate} \item $(34567, 987)$ \item $(560, 600)$ \item $(4554, 36)$ \item $(8098643070, 8173826342)$ \end{enumerate} \end{pro} \begin{pro} Solve the following linear diophantine equations, provided solutions exist:\begin{enumerate} \item $24x + 25y = 18$ \item $3456x + 246y = 44$ \item $1998x + 2000y = 33$ \end{enumerate} \end{pro} \begin{pro} Prove that the area of the triangle whose vertices are $(0, 0), (b, a), (x, y)$ is $$ \dfrac{|by - ax|}{2}.$$\end{pro} \begin{pro} A woman pays $\$ 2.78$ for some bananas and eggs. If each banana costs $\$ 0.69$ and each egg costs $\$ 0.35$, how many eggs and how many bananas did the woman buy?\end{pro} \end{multicols} \section{Linear Congruences} We recall that the expression $ax \equiv b\mod n$ means that there is $t \in \BBZ$ such that $ax = b + nt$. Hence, the congruencial equation in $x$, $ax \equiv b\mod n$ is soluble if and only if the linear diophantine equation $ax + ny = b$ is soluble. It is clear then that the congruence $$ ax \equiv b \ \mod \ n$$has a solution if and only if $(a, n)|b.$ \begin{thm}\label{thm:solutions_linear_congruence} Let $a$, $b$, $n$ be integers. If the congruence $ax \equiv b \ \mod \ n$ has a solution, then it has $(a, n)$ incongruent solutions $\mod n$.\end{thm} \begin{pf} From Theorem \ref{thm:solutions_linear_diophantine} we know that the solutions of the linear diophantine equation $ax + ny = b$ have the form $x = x_0 + nt/d, y = y_0 - at/d, d = (a, n), t \in \BBZ$, where $x_0, y_0$ satisfy $ax_0 + ny = b.$ Letting $t$ take on the values $t = 0, 1, \ldots ((a, n) - 1),$ we obtain $(a, n)$ mutually incongruent solutions, since the absolute difference between any two of them is less than $n.$ If $x = x_0 + nt'/d$ is any other solution, we write $t'$ as $t' = qd + r, 0 \leq r < d$. Then $$ \begin{array}{lcl} x & = & x_0 + n(qd + r)/d \\ & = & x_0 + nq + nr/d \\ & \equiv & x_0 + nr/d \ \mod \ n. \end{array}$$Thus every solution of the congruence $ax \equiv b\mod n$ is congruent $\mod n$ to one and only one of the $d$ values $x_0 + nt/d, 0 \leq t \leq d - 1.$ Thus if there is a solution to the congruence, then there are $d$ incongruent solutions $\mod n.$\end{pf} \begin{exa} Find all solutions to the congruence $5x \equiv 3 \mod 7$ \end{exa} Solution: Notice that according to Theorem \ref{thm:solutions_linear_congruence}, there should only be one solution $\mod 7$, as $(5, 7) = 1.$ We first solve the linear diophantine equation $5x + 7y = 1.$ By the Euclidean Algorithm $$ \begin{array}{lcl} 7 & = & 5\cdot 1 + 2 \\ 5 & = & 2\cdot 2 + 1 \\ 2 & = & 2 \cdot 1. \end{array}$$Hence, $$ \begin{array}{lcl} 1 & = & 5 - 2\cdot 2 \\ 2 & = & 7 - 5\cdot 1, \end{array}$$which gives $$ 1 = 5 - 2\cdot 2 = 5 - 2(7 - 5\cdot 1) = 5\cdot 3 - 7\cdot 2.$$ Whence $3 = 5(9) - 7(6).$ This gives $5\cdot 9 \equiv 3\mod 7$ which is the same as $5\cdot 2 \equiv 3\mod 7$. Thus $x \equiv 2\mod 7$. \begin{exa}Solve the congruence $$ 3x \equiv 6 \ \mod \ 12.$$ \end{exa} Solution: As $(3, 12) = 3$ and $3|6,$ the congruence has three mutually incongruent solutions. By inspection we see that $x = 2$ is a solution. By Theorem \ref{thm:solutions_linear_diophantine}, all the solutions are thus of the form $x = 2 + 4t, t \in \BBZ$. By letting $t = 0, 1, 2,$ the three incongruent solutions modulo $12$ are $t = 2, 6, 10$. \bigskip We now add a few theorems and definitions that will be of use in the future. \begin{thm}\label{thm:reducing_mod_linear_cong} Let $x$, $y$ be integers and let $a$, $n$ be non-zero integers. Then $$ ax \equiv ay \ \mod \ n $$if and only if $$ x \equiv y \ \mod \ \dfrac{n}{(a, n)}.$$ \end{thm} \begin{pf} If $ax \equiv ay\mod n$ then $a(x - y) = sn$ for some integer $s$. This yields $$ (x - y)\dfrac{a}{(a, n)} = s\dfrac{n}{(a, n)}.$$Since $(a/(a, n), n/(a, n)) = 1$ by Theorem \ref{thm:dividing_by_gcd}, we must have $$\dfrac{n}{(a, n)}|(x - y),$$ by Euclid's Lemma (Lemma \ref{lem:euclid}). This implies that $$x \equiv y \ \mod \ \dfrac{n}{(a, n)}.$$ Conversely if $x \equiv y \ \mod \ \dfrac{n}{(a, n)}$ implies $$ax \equiv ay \ \mod \ \dfrac{an}{(a, n)},$$ upon multiplying by $a.$ As $(a, n)$ divides $a$, the above congruence implies a fortiori that $ax - ay = tn$ for some integer $t$. This gives the required result.\end{pf} Theorem \ref{thm:reducing_mod_linear_cong} gives immediately the following corollary. \begin{cor} \label{cor:reducing_modulus}If $ax \equiv ay\mod n$ and $(a, n) = 1$, then $x \equiv y\mod n$. \end{cor} \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro} Solve the congruence $50x \equiv 12 \ \mod \ 14$.\end{pro} \begin{pro} How many $x$, $38 \leq x \leq 289$ satisfy $$ 3x \equiv 8 \ {\rm\mod \ 11}?$$\end{pro} \end{multicols} \section{A theorem of Frobenius} If $(a, b) = d > 1$ then the linear form $ax + by$ skips all non-multiples of $d.$ If $(a, b) = 1$, there is always an integer solution to $ax + by = n$ regardless of the integer $n.$ We will prove the following theorem of Frobenius that tells un when we will find nonnegative solutions to $ax + by = n.$ \begin{thm}[Frobenius]\label{thm:frobenius} Let $a, b$ be positive integers. If $(a, b) = 1$ then the number of positive integers m that cannot be written in the form $ar + bs = m$ for nonnegative integers r, s equals $(a - 1)(b - 1)/2$. \end{thm} \begin{pf} Let us say that an integer $n$ is {\em attainable} if there are nonnegative integers $r, s$ with $ar + bs = n$. Consider the infinite array $${\everymath{\displaystyle}\begin{array}{rrrrrrr} 0 & 1 & 2 & \ldots & k & \ldots & a - 1 \\ a & a + 1 & a + 2 & \ldots & a + k & \ldots & 2a - 1 \\ 2a & 2a + 1 & 2a + 2 & \ldots & 2a + k & \ldots & 3a - 1 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ \end{array}}$$ The columns of this array are arithmetic progressions with common difference $a$. The numbers directly below a number $n$ have the form $n + ka$ where $k$ is a natural number. Clearly, if $n$ is attainable, so is $n + ka$, implying thus that if an integer $n$ is attainable so is every integer directly below it. Clearly all multiples of $b$ are attainable. We claim that no two distinct multiples of $b, vb$ and $wb$ with $0 \leq v, w \leq a - 1$ can belong to the same column. If this were so then we would have $vb \equiv wb\mod a$. Hence $a(v - w) \equiv 0\mod a$. Since $(a, b) = 1$ we invoke Corollary 5.1 to deduce $v - w \equiv 0\mod a$. Since $0 \leq v, w \leq a - 1,$ we must have $v = w.$ \bigskip Now we show that any number directly above one of the multiples $vb, 0 \leq v \leq a - 1$ is non-attainable. For a number directly above $vb$ is of the form $vb - ka$ for some natural number $k$. If $vb - ka$ were attainable, then $ax + by = vb - ka$ for some nonnegative integers $x, y.$ This yields $by \leq ax + by = vb - ka < vb.$ Hence, $0 \leq y < v < a.$ This implies that $y \not\equiv v\mod b$. On the other hand, two numbers on the same column are congruent $\mod a$. Therefore we deduce $vb \equiv bv - ka \equiv ax + by\mod a$ which yields $bv \equiv by\mod a$. By Corollary \ref{cor:reducing_modulus} we obtain $v \equiv y\mod a$. This contradicts the fact that $0 \leq y < v < a.$ Thus the number of unattainable numbers is precisely the numbers that occur just above a number of the form $vb, 0 \leq v \leq a - 1.$ Now, on the $j$-th column, there are $(vb - j)/a$ values above $vb$. Hence the number of unattainable numbers is given by $$ \sum _{v = 0} ^{ a - 1} \sum _{j = 0} ^{a - 1} \dfrac{vb - j}{a} = \dfrac{(a - 1)(b - 1)}{2}, $$ as we wanted to show.\end{pf} \bigskip The greatest unattainable integer occurs just above $(a - 1)b$, hence the greatest value that is not attainable is $(a - 1)b - a$, which gives the following theorem. \bigskip \begin{thm} \label{thm:frobenius2} Let $a, b$ be relatively prime positive integers. Then the equation $$ ax + by = n$$ is unsoluble in nonnegative integers $x, y$ for $n = ab - a - b$. If $n > ab - a - b,$ then the equation is soluble in nonnegative integers.\end{thm} \begin{exa}[Putnam, 1971] A game of solitaire is played as follows. After each play, according to the outcome, the player receives either $a$ or $b$ points, ($a, b \in \BBN , a > b$), and his score accumulates from play to play. It has been noticed that there are thirty five non-attainable scores and that one of these is $58$. Find $a$ and $b$. \end{exa} Solution: The attainable scores are the nonnegative integers of the form $ax + by$. If $(a, b) > 1,$ there are infinitely many such integers. Hence $(a, b) = 1.$ By Theorem \ref{thm:frobenius}, the number of non-attainable scores is $(a - 1)(b - 1)/2$. Therefore, $(a - 1)(b - 1) = 70 = 2(35) = 5(14) = 7(10).$ The conditions $a > b, (a, b) = 1$ yield the two possibilities $a = 71, b = 2$ and $a = 11, b = 8$. As $58 = 0\cdot 71 + 2\cdot 29,$ the first alternative is dismissed. The line $11x + 8y = 58$ passes through $(6, -1)$ and $(-2, 10)$ and thus it does not pass through a lattice point in the first quadrant. The unique solution is $a = 11, b = 8$. \begin{exa}[AIME, 1994] Ninety-four bricks, each measuring $4''\times 10''\times 19''$, are to be stacked one on top of another to form a tower $94$ bricks tall. Each brick can be oriented so it contributes $4''$ or $10''$ or $19''$ to the total height of the tower. How many different tower heights can be achieved using all $94$ of the bricks? \end{exa} Solution: Let there be $x, y, z$ bricks of height $4'', 10'',$ and $19''$ respectively. We are asking for the number of different sums $$ 4x + 10y + 19z $$ with the constraints $x \geq 0, y \geq 0, z \geq 0, x + y + z = 94.$ \bigskip Now, $4x + 10y + 19z \leq 19\cdot 94 = 1786.$ Letting $x = 94 - y - z,$ we count the number of different nonnegative integral solutions to the inequality $376 + 3(2y + 5z) \leq 1786, y + z \leq 94,$ that is $2y + 5z \leq 470, y + z \leq 94.$ By Theorem \ref{thm:frobenius2}, every integer $\geq (2 - 1)(5 - 1) = 4$ can be written in the form $2y + 5z$, and the number of exceptions is $(2 - 1)(5 - 1)/2 = 2,$ namely $n = 1$ and $n = 3.$ Thus of the $471$ nonnegative integers $n \leq 470,$ we see that $469$ can be written in the form $n = 2y + 5z.$ Using $x = 96 - x - y$, $n, 4 \leq n \leq 470$ will be ``good'' only if we have $470 - n = 3x + 5z.$ By Theorem \ref{thm:frobenius} there are $(3 - 1)(5 - 1)/2 = 4$ exceptions, each $\leq 8,$ namely $n = 1, 2, 4, 7.$ This means that 463, 466, 468, and 469 are not representable in the form $4x + 10y + 19z.$ Then every integer $n, 0 \leq n \leq 470$ except for 1, 3, 463, 466, 468, and 469 can be thus represented, and the number of different sums is $471 - 6 = 465.$ \begin{exa} \begin{enumerate}\item Let $(n, 1991) = 1$. Prove that $\dfrac{n}{1991}$ is the sum of two positive integers with denominator $< 1991$ if an only if there exist integers $m, a, b$ with $$ (*) \ \ \,\,\,\,\, 1 \leq m \leq 10, \ a \geq 1, \ b \geq 1, \ mn = 11a + 181b .$$ \item Find the largest positive rational with denominator $1991$ that cannot be written as the sum of two positive rationals each with denominators less than $1991$. \end{enumerate}\end{exa} Solution: (a) If $(*)$ holds then $\dfrac{n}{1991} = \dfrac{a}{181m} + \dfrac{b}{11m}$ does the trick. Conversely, if $\dfrac{n}{1991} = \dfrac{a}{r} + \dfrac{b}{s}$ for $a, b \geq 1, (a, r) = (b, s) = 1,$ and $r, s < 1991$, we may suppose $r = 181r_1 , s = 11s_1$ and then $nr_1 s_1 = 11as_1 + 181br_1$, which leads to $r_1 |11as_1$ and so $r_1 |s_1$. Similarly, $s_1 |r_1$, whence $r_1 = s_1 = m,$ say, and $(*)$ follows. \\ (b) Any $n > 170, (n, 1991) = 1$ satisfies $(*)$ with $b = 1$ and ${\mathscr M}$ such that $mn$ is of the form $mn \equiv 181\mod 11$. For $mn > 181$ except if $m = 1, n \leq 180;$ but then $n$ would not be of the form $n \equiv 181\mod 11$. \bigskip But $n = 170$ does not satisfy $(*)$; for we would have $170 \equiv 181b\mod 11$, so $b \equiv m\mod 11$, which yields $b \geq m,$ but $170m < 181$. The answer is thus $170/1991$. \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro} Let $a, b, c$ be positive real numbers. Prove that there are at least ${c^2}/{2ab}$ pairs of integers $(x, y)$ satisfying $$ x \geq 0, \, y \geq 0, \,\, ax + by \leq c.$$\end{pro} \begin{pro}[AIME, 1995] What is largest positive integer that is not the sum of a positive integral multiple of $42$ and a positive composite integer?\end{pro} \begin{pro} Let $a > 0, b > 0, (a, b) = 1.$ Then the number of nonnegative solutions to the equation $ax + by = n$ is equal to $$ [\dfrac{n}{ab}] \ {\rm or} \ [\dfrac{n}{ab}] + 1.$$ (Hint: $[s] - [t] = [s - t]$ or $[s - t] + 1.$)\end{pro} \begin{pro}Let $a, b \in \BBN , (a, b) = 1$. Let $S(n)$ denote the number of nonnegative solutions to $$ ax + by = n.$$ Evaluate $$ \lim _{n\rightarrow\infty} \ \dfrac{S(n)}{n}.$$\end{pro} \begin{pro}[IMO, 1983] Let $a, b, c$ be pairwise relatively prime integers. Demonstrate that $2abc - ab - bc - ca$ is the largest integer not of the form $$ bcx + acy + abz, \ {\hspace{5mm}} x \geq 0, y \geq 0, z \geq 0.$$\end{pro} \end{multicols} \section{Chinese Remainder Theorem} In this section we consider the case when we have multiple congruences. Consider the following problem: find an integer $x$ which leaves remainder 2 when divided by 5, is divisible by 7, and leaves remainder 4 when divided by 11. In the language of congruences we are seeking $x$ such that $$ \begin{array}{llll} x & \equiv & 2 & {\rm\mod \ 5,} \\ x & \equiv & 0 & {\rm\mod \ 7,} \\ x & \equiv & 4 & {\rm\mod \ 11.} \end{array}$$ One may check that $x = 147$ satisfies the requirements, and that in fact, so does the parametric family $x = 147 + 385t, t \in \BBZ$. \bigskip We will develop a method to solve congruences like this one. The method is credited to the ancient Chinese, and it is thus called the {\em Chinese Remainder Theorem.} \bigskip \begin{exa} Find x such that $$ x \equiv 3 \ {\rm\mod \ 5 \ and} \ x \equiv 7 \ {\rm\mod \ 11.}$$\end{exa} Solution: Since $x = 3 + 5a,$ we have $11x = 33 + 55a.$ As $x = 7 + 11b,$ we have $5x = 35 + 55b.$ Thus $x = 11x - 10x = 33 - 70 + 55a - 110b.$ This means that $x \equiv -37 \equiv 18\mod 55$. One verifies that all the numbers $x = 18 + 55t, t\in \BBZ$ verify the given congruences. \begin{exa}Find a number n such that when divided by $4$ leaves remainder $2$, when divided by $5$ leaves remainder $1$, and when divided by $7$ leaves remainder $1$. \end{exa} Solution: We want $n$ such that $$\begin{array}{lll} n \equiv & 2 & \mod \ 4, \\ n \equiv & 1 & \mod \ 5, \\ n \equiv & 1 & \mod \ 7. \end{array} $$ This implies that $$\begin{array}{lll} 35n \equiv & 70 & \mod \ 140, \\ 28n \equiv & 28 & \mod \ 140, \\ 20n \equiv & 20 & \mod \ 140. \end{array} $$ \bigskip As $n = 21n - 20n$, we have $n \equiv 3(35n - 28n) - 20n \equiv 3(70 - 28) - 20 \equiv 106\mod 140$. Thus all $n \equiv 106\mod 140$ will do. \begin{thm}[Chinese Remainder Theorem]\label{thm:chinese_remainder} Let $m_1, m_2, \ldots m_k$ be pairwise relatively prime positive integers, each exceeding $1,$ and let $a_1, a_2, \ldots a_k$ be arbitrary integers. Then the system of congruences $$ \begin{array}{llll} x & \equiv & a_1 & \mod \ m_1 \\ x & \equiv & a_2 & \mod \ m_2 \\ \vdots & \vdots & \vdots & \\ x & \equiv & a_k & \mod \ m_k \end{array}$$has a unique solution modulo $m_1m_2\cdots m_k.$ \end{thm} \begin{pf} Set $P_j = m_1m_2\cdots m_k/m_j, 1 \leq j \leq k.$ Let $Q_j$ be the inverse of $P_j\mod m_j$, i.e., $P_jQ_j \equiv 1 \mod m_j$, which we know exists since all the $m_i$ are pairwise relatively prime. Form the number $$ x = a_1P_1Q_1 + a_2P_2Q_2 + \cdots + a_kP_kQ_k.$$ This number clearly satisfies the conditions of the theorem. The uniqueness of the solution modulo $m_1m_2 \cdots m_k$ can be easily established. \end{pf} \begin{exa} Can one find one million consecutive integers that are not square-free?\end{exa} Solution: Yes. Let $p_1, p_2, \ldots , p_{1000000}$ be a million different primes. By the Chinese Remainder Theorem, there exists a solution to the following system of congruences. $$ \begin{array}{llcl} x & \equiv & -1 & \mod \ p_1 ^2, \\ x & \equiv & -2 & \mod \ p_2 ^2, \\ \vdots & \vdots & \vdots & \vdots \\ x & \equiv & -1000000 & \mod \ p_{1000000} ^2. \end{array}$$The numbers $x + 1, x + 2, \ldots , x + 1000000$ are a million consecutive integers, each of which is divisible by the square of a prime. \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro} Solve the following systems: \begin{enumerate} \item $x \equiv -1 \mod 4$; $x \equiv 2 \mod 5$ \item $4x \equiv 3 \mod 7$; $x \equiv 10 \mod 11$ \item $5x \equiv 2 \mod 8$; $3x \equiv 2 \mod 9$; $x \equiv 0 \mod 11$ \end{enumerate}\end{pro} \begin{pro}[USAMO 1986] \begin{enumerate} \item Do there exist fourteen consecutive positive integers each of which is divisible by one or more primes $p, 2 \leq p \leq 11$? \item Do there exist twenty-one consecutive integers each of which is divisible by one or more primes $p, 2 \leq p \leq 13$?\end{enumerate}\end{pro} \end{multicols} \chapter{Number-Theoretic Functions} \section{Greatest Integer Function} The largest integer not exceeding $x$ is denoted by $\floor{x}$ or $\floor{ x }$. We also call this function the {\em floor} function. Thus $\floor{x}$ satisfies the inequalities $x - 1 < \floor{x} \leq x,$ which, of course, can also be written as $\floor{x} \leq x < \floor{x} + 1.$ The fact that $\floor{x}$ is the {\em unique} integer satisfying these inequalities, is often of use. We also utilise the notation $\{ x\} = x - \floor{x},$ to denote the fractional part of $x$, and $||x|| = \min _{n \in \BBZ } |x - n|$ to denote the distance of a real number to its nearest integer. A useful fact is that we can write any real number $x$ in the form $x = \floor{x} + \{ x\}, 0 \leq \{ x \} < 1.$ The greatest integer function enjoys the following properties: \begin{thm} \label{thm:floor_properties}Let $\alpha , \beta \in \BBR , a \in \BBZ, n \in \BBN$. Then \begin{enumerate} \item $\floor{\alpha + a} = \floor{\alpha } + a$ \item $\floor{\dfrac{\alpha }{n}} =\floor{\dfrac{\floor{\alpha }}{n}}$ \item $\floor{\alpha }+ \floor{\beta } \leq \floor{\alpha + \beta } \leq \floor{\alpha } + \floor{\beta } + 1$ \end{enumerate} \end{thm} \begin{pf} \begin{enumerate} \item Let $m = \floor{\alpha + a}$. Then $m \leq \alpha + a < m + 1.$ Hence $m - a \leq \alpha < m - a + 1.$ This means that $m - a = \floor{\alpha}$, which is what we wanted. \item Write ${\alpha}/{n}$ as ${\alpha}/{n} = \floor{{\alpha}/{n}} + \theta , 0 \leq \theta < 1.$ Since $n\floor{{\alpha}/{n}}$ is an integer, we deduce by (1) that $$ \floor{\alpha } = \floor{n\floor{\alpha /n} + n\theta} = n\floor{\alpha /n} + \floor{n\theta}.$$ Now, $0 \leq \floor{n\theta} \leq n\theta < n,$ and so $0 \leq \floor{n\theta }/n < 1.$ If we let $\Theta = \floor{n\theta }/n,$ we obtain $$ \dfrac{\floor{\alpha }}{n} = \floor{\dfrac{\alpha}{n}} + \Theta , \ \ 0 \leq \Theta < 1.$$ This yields the required result. \item From the inequalities $\alpha - 1 < \floor{\alpha } \leq \alpha, \beta - 1 < \floor{\beta } \leq \beta$ we get $\alpha + \beta - 2 < \floor{\alpha }+ \floor{\beta } \leq \alpha + \beta $. Since $\floor{\alpha } + \floor{\beta }$ is an integer less than or equal to $\alpha + \beta ,$ it must be less than or equal to the integral part of $\alpha + \beta ,$ i.e. $\floor{\alpha + \beta }.$ We obtain thus $\floor{\alpha } + \floor{\beta } \leq \floor{\alpha + \beta }.$ Also, $\alpha + \beta$ is less than the integer $\floor{\alpha } + \floor{\beta } + 2$, so its integer part $\floor{\alpha + \beta }$ must be less than $\floor{\alpha }+ \floor{\beta} + 2$, but $\floor{\alpha + \beta } < \floor{\alpha } + \floor{\beta } + 2$ yields $\floor{\alpha + \beta } \leq \floor{\alpha } + \floor{\beta } + 1$. This proves the inequalities. \end{enumerate} \end{pf} \begin{exa} Find a non-zero polynomial $P(x, y)$ such that $$P(\floor{2t}, \floor{3t}) = 0$$ for all real $t$. \end{exa} Solution: We claim that $3[2t] - 2[3t] = 0, \pm 1$ or $-2.$ We can then take $$P(x, y) = (3x - 2y)(3x - 2y - 1)(3x - 2y + 1)(3x - 2y + 2).$$ In order to prove the claim, we observe that $\floor{x}$ has unit period, so it is enough to prove the claim for $t \in [0, 1)$. We divide $[0, 1)$ as $$ [0, 1) = [0, 1/3) \cup [1/3, 1/2) \cup [1/2, 2/3) \cup [2/3, 1).$$ If $t\in [0, 1/3),$ then both $\floor{2t}$ and $\floor{3t}$ are $= 0,$ and so $3\floor{2t} - 2\floor{3t} = 0.$ If $t\in [1/3, 1/2)$ then $[3t] = 1$ and $[2t] = 0,$ and so $3\floor{2t} - 2\floor{3t} = - 2$. If $t\in [1/2, 2/3),$ then $[2t] = 1, [3t] = 1$, and so $3\floor{2t} - 2\floor{3t} = 1.$ If $t\in [2/3 , 1)$, then $\floor{2t} = 1, \floor{3t} = 2,$ and $3\floor{2t} - 2\floor{3t} = -1.$ \begin{exa} Describe all integers $n$ such that $1 + \floor{\sqrt{2n}}\Big|2n.$ \end{exa}Solution: Let $2n = m(1 + \floor{\sqrt{2n}})$. If $m \leq \floor{\sqrt{2n}} - 1$ then $2n \leq (\floor{\sqrt{2n}} - 1)(\floor{\sqrt{2n}} + 1) = \floor{\sqrt{2n}}^2 - 1 \leq 2n - 1 < 2n,$ a contradiction. If $m \geq \floor{\sqrt{2n}} + 1,$ then $2n \geq (\floor{\sqrt{2n}}^2 + 1)^2 \geq 2n + 1,$ another contradiction. It must be the case that $m = \floor{\sqrt{2n}}$. \bigskip Conversely, let $n = \dfrac{l(l + 1)}{2}$. Since $l < \sqrt{2n} < l + 1, l = \floor{\sqrt{2n}}$. So all the integers with the required property are the triangular numbers. \begin{exa} Prove that the integers $$ \floor{ \left( 1 + \sqrt{2}\right)^n}$$ with $n$ a nonnegative integer, are alternately even or odd. \end{exa} Solution: By the Binomial Theorem $$ (1 + \sqrt{2})^{n} + (1 - \sqrt{2})^n = 2 \sum _{0 \leq k \leq n/2} (2)^k \binom{n}{2k} := 2N,$$an even integer. Since $-1 < 1 - \sqrt{2} < 0$, it must be the case that $(1 - \sqrt{2})^n$ is the fractional part of $(1 + \sqrt{2})^n$ or $(1 + \sqrt{2})^n + 1$ depending on whether $n$ is odd or even, respectively. Thus for odd $n$, $(1 + \sqrt{2})^n - 1 < (1 + \sqrt{2})^n + (1 - \sqrt{2})^n < (1 + \sqrt{2})^n $, whence $(1 + \sqrt{2})^n + (1 - \sqrt{2})^n = \floor{(1 + \sqrt{2})^n}$, always even, and for $n$ even $2N := (1 + \sqrt{2})^n + (1 - \sqrt{2})^n = \floor{(1 + \sqrt{2})^n} + 1$, and so $\floor{(1 + \sqrt{2})^n} = 2N - 1,$ always odd for even $n.$ \begin{exa} Prove that the first thousand digits after the decimal point in $$ (6 + \sqrt{35})^{1980}$$ are all $9$'s. \end{exa} Solution: Reasoning as in the preceding problem, $$ (6 + \sqrt{35})^{1980} + (6 - \sqrt{35})^{1980} = 2k,$$an even integer. But $0 < 6 - \sqrt{35} < 1/10$, (for if $\dfrac{1}{10} < 6 - \sqrt{35}$, upon squaring $3500 < 3481$, which is clearly nonsense), and hence $ 0 < (6 - \sqrt{35})^{1980} < 10^{-1980} $ which yields $$2k - 1 + \underbrace{0.9\ldots 9}_{1979 \ {\rm nines}} = 2k - \dfrac{1}{10^{1980}} < (6 + \sqrt{35})^{1980} < 2k,$$ This proves the assertion of the problem. \begin{exa}[Putnam 1948] If $n$ is a positive integer, demonstrate that $$ \floor{ \sqrt{n} + \sqrt{n + 1} } = \floor{ \sqrt{4n + 2} } .$$ \end{exa} Solution: By squaring, it is easy to see that $$ \sqrt{4n + 1} < \sqrt{n} + \sqrt{n + 1} < \sqrt{4n + 3}.$$ Neither $4n + 2$ nor $4n + 3$ are squares since squares are either congruent to $0$ or $1$ mod $4$, so $$ \floor{ \sqrt{4n + 2}} = \floor{ \sqrt{4n + 3}},$$ and the result follows. \begin{exa} Find a formula for the $n$-th non-square. \end{exa} Solution: Let $T_n$ be the $n$-th non-square. There is a natural number $m$ such that $m^2 < T_n < (m + 1)^2$. As there are $m$ squares less than $T_n$ and $n$ non-squares up to $T_n$, we see that $T_n = n + m.$ We have then $m^2 < n + m < (m + 1)^2$ or $m^2 - m < n < m^2 + m + 1.$ Since $n, m^2 - m, m^2 + m + 1$ are all integers, these inequalities imply $m^2 - m + \dfrac{1}{4} < n < m^2 + m + \dfrac{1}{4}$, that is to say, $(m - 1/2)^2 < n < (m + 1/2)^2.$ But then $m = \floor{ \sqrt{n} + \dfrac{1}{2}}.$ Thus the $n$-th non-square is $T_n = n + \floor{ \sqrt{n} + 1/2}.$ \begin{exa}[Putnam 1983] Let $f(n) = n + \floor{ \sqrt{n}}$. Prove that for every positive integer m, the sequence $$ m, f(m), f(f(m)), f(f(f(m))), \ldots$$contains at least one square of an integer.\end{exa} Solution: Let $m = k^2 + j, 0 \leq j \leq 2k$. Split the $m$'s into two sets, the set $A$ of all the $m$ with excess $j, 0 \leq j \leq k$ and the set $B$ with all those $m$'s with excess $j, k < j < 2k + 1.$ Observe that $k^2 \leq m < (k + 1)^2 = k^2 + 2k + 1.$ If $j = 0$, we have nothing to prove. Assume that $m \in B.$ As $\floor{ \sqrt{m}} = k$, $f(m) = k^2 + j + k = (k + 1)^2 + j - k - 1,$ with $0 \leq j - k - 1 \leq k - 1 < k + 1.$ This means that either $f(m)$ is a square or $f(m) \in A.$ It is thus enough to consider the alternative $m \in A$, in which case $\floor{ \sqrt{m + k}} = k$ and $$f(f(m)) = f(m + k) = m + 2k = (k + 1)^2 + j - 1.$$This means that $f(f(m))$ is either a square or $f(f(m)) \in A$ with an excess $j - 1$ smaller than the excess $j$ of $m.$ At each iteration the excess will reduce and eventually it will hit $0$, whence we reach a square. \begin{exa} Solve the equation $$ \floor{ x^2 - x - 2} = \floor{x},$$ for $x\in \BBR$.\end{exa} Solution: Observe that $\floor{ a} = \floor{ b}$ if and only if $\exists k \in \BBZ$ with $a, b \in [k, k + 1)$ which happens if and only if $|a - b| < 1.$ Hence, the given equation has a solution if and only if $|x^2 - 2x - 2| < 1$. Solving these inequalities it is easy to see that the solution is thus $$ x\in (-1, \dfrac{1}{2}(1 - \sqrt{5})]\cup [\dfrac{1}{2}(1 + \sqrt{17}), \dfrac{1}{2}(1 + \sqrt{21})).$$ \begin{thm}\label{thm:lattice_points} If $a, b$ are relatively prime natural numbers then $$ \sum _{k = 1} ^{a - 1} \floor{ \dfrac{kb}{a} } = \sum _{k = 1} ^{b - 1} \floor{ \dfrac{ka}{b} }= \dfrac{(a - 1)(b - 1)}{2}.$$ \end{thm}\begin{pf} Consider the rectangle with vertices at $(0, 0), (0, b), (a, 0), (a, b)$. This rectangle contains $(a - 1)(b - 1)$ lattice points, i.e., points with integer coordinates. This rectangle is split into two halves by the line $y = \dfrac{xb}{a}$. We claim that there are no lattice points on this line, except for the endpoints. For if there were a lattice point $(m, n), 0 < m < a, 0 < n < b,$ then $\dfrac{n}{m} = \dfrac{b}{a}$. Thus $n/m$ is a reduction for the irreducible fraction $b/a,$ a contradiction. The points $L_k = (k, \dfrac{kb}{a}), 1 \leq k \leq a - 1$ are each on this line. Now, $\floor{ \dfrac{kb}{a}}$ equals the number of lattice points on the vertical line that goes from $(k, 0)$ to $(k, \dfrac{kb}{a})$, i.e. $\sum _{k = 1} ^{a - 1} \floor{ \dfrac{kb}{a} }$ is the number of lattice points on the lower half of the rectangle. Similarly, $\sum _{k = 1} ^{b - 1} \floor{ \dfrac{ka}{b} }$ equals the number of lattice points on the upper half of the rectangle. Since there are $(a - 1)(b - 1)$ lattice points in total, and their number is shared equally by the halves, the assertion follows. \end{pf} \begin{exa} Find the integral part of $$ \sum _{k = 1} ^{10^6} \dfrac{1}{\sqrt{k}}.$$\end{exa} Solution: The function $x \mapsto x^{-1/2}$ is decreasing. Thus for positive integer $k,$ $$ \dfrac{1}{\sqrt{k + 1}} < \int _k ^{k + 1} \dfrac{dx}{\sqrt{x}} < \dfrac{1}{\sqrt{k}}.$$ Summing from $k = 1$ to $k = 10^6 - 1$ we deduce $$ \sum _{k = 2} ^{10^6} \dfrac{1}{\sqrt{k}} < \int _{1} ^{10^6} \dfrac{dx}{\sqrt{x}} < \sum _{k = 1} ^{10^6 - 1}\dfrac{1}{\sqrt{k}}.$$ The integral is easily seen to be $1998$. Hence $$ 1998 + 1/10^3 < \sum _{k = 1} ^{10^6} \dfrac{1}{\sqrt{k}} < 1999 .$$ The integral part sought is thus $1998$. \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro} Prove that for all real numbers $x, y$, $$ \floor{x} + \floor{ x + y} + \floor{ y} \leq \floor{ 2x} + \floor{ 2y}$$holds. \end{pro} \begin{pro} If $x$, $y$ real numbers, when is it true that $\floor{x}\floor{ y} \leq \floor{ xy}$?\end{pro} \begin{pro} If $n > 1$ is a natural number and $\alpha \geq 1$ is a real number, prove that $$ [\alpha ] > \floor{ \dfrac{\alpha}{n} }.$$ \end{pro} \begin{pro} If $a$, $b$, $n$ are positive integers, prove that $$ \floor{ \dfrac{ab}{n} } \geq a\floor{ \dfrac{b}{n} }.$$ \end{pro} \begin{pro} Let $\alpha$ be a real number. Prove that $[\alpha ] + [-\alpha ] = -1$ or $0$ and that $\floor{ \alpha } - 2\floor{ \alpha /2} = 0$ or $1$.\end{pro} \begin{pro} Prove that $$\floor{ (2 + \sqrt{3})^n }$$is an odd integer. \end{pro} \begin{pro} Show that the $n$-th element of the sequence $$1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, \ldots$$where there are $n$ occurrences of the integer $n$ is $\floor{ \sqrt{2n} + 1/2}$. \end{pro} \begin{pro} Prove {\em Hermite's Identity}: if $x$ is a real number and $n$ is a natural number then $$ \floor{ nx } = \floor{ x } + \floor{ x + \dfrac{1}{n} } + \floor{ x + \dfrac{2}{n} } + \cdots + \floor{ x + \dfrac{n - 1}{n} } . $$ \end{pro} \begin{pro} Prove that for all integers $m$, $n$, the equality $$ \floor{ \dfrac{m + n}{2} } + \floor{ \dfrac{n - m + 1}{2} } = n$$ holds.\end{pro} \begin{pro} If $a$, $b$, $c$, $d$ are positive real numbers such that $$ \floor{ na} + \floor{ nb} = \floor{ nc} + \floor{ nd}$$ for all natural numbers $n$, prove that $$a + b = c + d.$$\end{pro} \begin{pro} If $n$ is a natural number, prove that $$ \floor{ \dfrac{n + 2 - \floor{ n/25}}{3} } = \floor{ \dfrac{8n + 24}{25} } .$$\end{pro} \begin{pro} Solve the equation $$ \floor{ \dfrac{x}{1994} } = \floor{ \dfrac{x}{1995} } .$$\end{pro} \begin{pro} Let $[\alpha , \beta]$ be an interval which contains no integers. Prove that there is a positive integer $n$ such that $[n\alpha , n\beta]$ still contains no integers but has length at least $1/6$. \end{pro} \begin{pro}[IMO 1968] For every natural number $n$, evaluate the sum $$\sum _{k = 0} ^{\infty} \floor{ \dfrac{n + 2^k}{2^{k + 1}} } .$$ \end{pro} \begin{pro}[Putnam 1973] Prove that if $n \in \BBN$, $$ \min _{k \in \BBN} (k + \floor{ n/k}) = \floor{ \sqrt{4n + 1}}. $$ \end{pro} \begin{pro}[Dirichlet's principle of the hyperbola] Let $N$ be the number of integer solutions to $xy \leq n, x > 0, y > 0.$ Prove that $$ N = \sum _{k = 1} ^n \floor{ \dfrac{n}{k} } = 2\sum _{1 \leq k \leq \sqrt{n}} \ \floor{ \dfrac{n}{k} } - \floor{ \sqrt{n}}^2 . $$\end{pro} \begin{pro}[Circle Problem] Let $r > 0$ and let $T$ denote the number of lattice points of the domain $x^2 + y^2 \leq r^2 .$ Prove that $$ T = 1 + 4\floor{ r} + 8\sum _{0 < x \leq r \sqrt{2}} \, \floor{ \sqrt{r^2 - x^2}} + 4\floor{ \dfrac{r}{\sqrt{2}} }^2 .$$\end{pro} \begin{pro}Let $d = (a, b)$. Prove that $$ \sum _{1 \leq n \leq b - 1} \, \floor{ \dfrac{an}{b} } = \dfrac{(a - 1)(b - 1)}{2} + \dfrac{d - 1}{2}.$$\end{pro} \begin{pro}[Eisenstein] If $(a, b) = 1$ and $a, b$ are odd, then $$ \sum _{1 \leq n \leq (b - 1)/2} \, \floor{ \dfrac{an}{b} } + \sum _{1 \leq n \leq (a - 1)/2} \, \floor{ \dfrac{bn}{a} } = \dfrac{(a - 1)(b - 1)}{4}. $$\end{pro} \begin{pro} Let $m\in \BBN$ with $m > 1$ and let $y$ be a positive real number. Prove that $$ \sum _x \floor{ \sqrt[m]{\dfrac{y}{x}} } = \floor{ y},$$where the summation runs through all positive integers $x$ not divisible by the $m$th power of an integer exceeding $1$.\end{pro} \begin{pro} For which natural numbers $n$ will $112$ divide $$ 4^n - \floor{ (2 + \sqrt{2})^n} ?$$\end{pro} \begin{pro} A {\em triangular number} is a number of the form $1 + 2 + \cdots + n, n \in \BBN$. Find a formula for the $n$th non-triangular number. \end{pro} \begin{pro}[AIME 1985] How many of the first thousand positive integers can be expressed in the form $$ \floor{ 2x} + \floor{ 4x} + \floor{ 6x} + \floor{ 8x}? $$\end{pro} \begin{pro}[AIME 1987] What is the largest positive integer $n$ for which there is a unique integer $k$ such that $$ \dfrac{8}{15} < \dfrac{n}{n + k} < \dfrac{7}{13} ?$$ \end{pro} \begin{pro} Prove that if $p$ is an odd prime, then $$ \floor{ (2 + \sqrt{5})^p } - 2^{p + 1}$$ is divisible by $p$. \end{pro} \begin{pro} Prove that the $n$-th number not of the form $\floor{ e^k }, k = 1, 2, \ldots$ is $$T_n = n + \floor{ \ln (n + 1 + \floor{ \ln (n + 1)})}.$$ \end{pro} \begin{pro}[Leningrad Olympiad] How many different integers are there in the sequence $$ \floor{ \dfrac{1^2}{1980} }, \floor{ \dfrac{2^2}{1980} }, \ldots , \floor{ \dfrac{1980^2}{1980} }?$$\end{pro} \begin{pro} Let $k \geq 2$ be a natural number and x a positive real number. Prove that $$ \floor{ \sqrt[k]{x} } = \floor{ \sqrt[k]{\floor{x}} } .$$\end{pro} \begin{pro}\begin{enumerate} \item Find a real number $x \neq 0$ such that $x, 2x, \ldots , 34x$ have no $7$'s in their decimal expansions. \item Prove that for any real number $x \neq 0$ at least one of $x , 2x, \ldots 79x$ has a $7$ in its decimal expansion. \item Can you improve the ``gap'' between $34$ and $79$? \end{enumerate}\end{pro} \begin{pro}[AIME 1991] Suppose that $r$ is a real number for which $$ \sum _{k = 19} ^{91} \floor{ r + \dfrac{k}{100} } = 546.$$ Find the value of $\floor{ 100r}$. \end{pro} \begin{pro}[AIME 1995] Let $f(n)$ denote the integer closest to $n^{1/4}$, when $n$ is a natural number. Find the exact numerical value of $$ \sum _{n = 1} ^{1995} \dfrac{1}{f(n)}.$$ \end{pro} \begin{pro} Prove that $$ \int _0 ^1 (-1)^{\floor{ 1994x} + \floor{ 1995x}}\binom{1993}{\floor{ 1994x}}\binom{1994}{\floor{ 1995x}}\, dx = 0.$$ \end{pro} \begin{pro} Prove that $$ \floor{ \sqrt{n} + \sqrt{n + 1} } = \floor{ \sqrt{n} + \sqrt{n + 2} } .$$\end{pro} \begin{pro}[Putnam 1976] Prove that $$ \lim _{n \rightarrow \infty} \sum _{1 \leq k \leq n} \left(\floor{ \dfrac{2n}{k} } - 2\floor{ \dfrac{n}{k} }\right) = \ln 4 - 1 . $$ \end{pro} \begin{pro}[Putnam 1983] Prove that $$ \lim _{n\rightarrow\infty} \dfrac{1}{n}\int _1 ^n \left|\left|\dfrac{n}{x}\right|\right| \, dx = \log_3 (4/\pi ).$$\end{pro} You may appeal to {\em Wallis Product Formula:} $$ \dfrac{2}{1}\cdot\dfrac{2}{3}\cdot\dfrac{4}{3}\cdot\dfrac{4}{5}\cdot\dfrac{6}{5}\cdot\dfrac{6}{7}\cdot\dfrac{8}{7}\cdot\dfrac{8}{9}\cdots = \dfrac{\pi}{2}.$$ \end{multicols} \section{De Polignac's Formula} We will consider now the following result due to De Polignac. \begin{thm}[De Polignac's Formula]\label{thm:depolignac} The highest power of a prime $p$ dividing $n!$ is given by $$\sum _{k = 1} ^{\infty} \floor{ \dfrac{n}{p^k} } .$$ \end{thm} \begin{pf} The number of integers contributing a factor of $p$ is $\floor{ n/p}$, the number of factors contributing a second factor of $p$ is $\floor{ n/p^2}$, etc..\end{pf} \begin{exa} How many zeroes are at the end of $300!$? \end{exa} Solution: The number of zeroes is determined by how many times $10$ divides into $300$. Since there are more factors of $2$ in $300!$ than factors of $5$, the number of zeroes is thus determined by the highest power of $5$ in $300!$. By De Polignac's Formula this is $\sum _{k = 1} ^\infty \floor{ 300/5^k} = 60 + 12 + 2 = 74.$ \begin{exa}Does $$ 7\Big|\binom{1000}{500}?$$\end{exa} Solution: The highest power of $7$ dividing into $1000! $ is $ \floor{ 1000/7} + \floor{ 1000/7^2} + \floor{ 1000/7^3} = 142 + 20 + 2 = 164.$ Similarly, the highest power of $7$ dividing into 500! is $71 + 10 + 1 = 82.$ Since $\binom{1000}{500} = \dfrac{1000!}{(500!)^2},$ the highest power of $7$ that divides $\binom{1000}{500}$ is $164 - 2\cdot 82 = 0,$ and so $7$ does not divide $\binom{1000}{500}.$ \begin{exa} Let $n = n_1 + n_2 + \cdots + n_k$ where the $n_i$ are nonnegative integers. Prove that the quantity $$ \dfrac{n!}{n_1 !n_2 ! \cdots n_k !}$$ is an integer. \end{exa}Solution: From (3) in Theorem \ref{thm:floor_properties} we deduce by induction that $$ \floor{ a_1} + \floor{ a_2} + \cdots + \floor{ a_l} \leq \floor{ a_1 + a_2 + \cdots + a_l}.$$For any prime $p$, the power of $p$ dividing $n!$ is $$ \sum _{j \geq 1} \floor{ n/p^j} = \sum _{j \geq 1} \floor{ (n_1 + n_2 + \cdots + n_k)/p^j}.$$The power of $p$ dividing $n_1!n_2!\cdots n_k!$ is $$\sum _{j \geq 1} \floor{ n_1/p^j} + \floor{ n_2/p^j} + \cdots \floor{ n_k/p^j}.$$Since $$ \floor{ n_1/p^j} + \floor{ n_2/p^j} + \cdots + \floor{ n_k/p^j} \leq \floor{ (n_1 + n_2 + \cdots + n_k)/p^j},$$ we see that the power of any prime dividing the numerator of $$ \dfrac{n!}{n_1!n_2!\cdots n_k!}$$is at least the power of the same prime dividing the denominator, which establishes the assertion. \begin{exa} Given a positive integer $n > 3,$ prove that the least common multiple of the products $x_1 x_2 \cdots x_k (k \geq 1),$ whose factors $x_i$ are the positive integers with $$ x_1 + x_2 + \cdots x_k \leq n,$$is less than $n!.$\end{exa} Solution: We claim that the least common multiple of the numbers in question is $$ \prod _{\stackrel{p}{p \ {\rm prime}}} p^{\floor{ n/p}}.$$Consider an arbitrary product $x_1x_2\cdots x_k,$ and an arbitrary prime $p.$ Suppose that $p^{\alpha _j}|x_j, p^{\alpha _j + 1}\not |x_j.$ Clearly $p^{\alpha _1} + \cdots + p{\alpha _k} \leq n$ and since $p^\alpha \geq \alpha p,$ we have $$ p(\alpha _1 + \cdots \alpha _k) \leq n \ {\rm or} \ \alpha _1 + \cdots + \alpha _k \leq \floor{ \dfrac{n}{p}}.$$Hence it follows that the exponent of an arbitrary prime $p$ is at most $\floor{ n/p}$. But on choosing $x_1 = \cdots = x_k = p, k = \floor{ n/p},$ we see that there is at least one product for which equality is achieved. This proves the claim. \bigskip The assertion of the problem now follows upon applying De Polignac's Formula and the claim. \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro}[AHSME 1977] Find the largest possible $n$ such that $10^n$ divides $1005!.$\end{pro} \begin{pro} Find the highest power of $17$ that divides $(17^n - 2)!$ for a positive integer $n$. \end{pro} \begin{pro} Find the exponent of the highest power of $24$ that divides $300!$. \end{pro} \begin{pro} Find the largest power of $7$ in $300!$. \end{pro} \begin{pro}[AIME 1983] What is the largest two-digit prime factor of the integer $$ \binom{200}{100}?$$ \end{pro} \begin{pro}[USAMO 1975] \begin{enumerate} \item Prove that $$ \floor{ 5x} + \floor{ 5y} \geq \floor{ 3x + y} + \floor{ 3y + x}.$$ \item Using the result of part 1 or otherwise, prove that $$ \dfrac{(5m)!(5n)!}{m!n!(3m + n)!(3n + m)!}$$is an integer for all positive integers $m, n.$\end{enumerate}\end{pro} \begin{pro} Prove that if $n > 1, (n, 6) = 1$, then $$ \dfrac{(2n - 4)!}{n!(n - 2)!}$$is an integer. \end{pro} \begin{pro}[AIME 1992] Define a positive integer n to be a ``factorial tail'' if there is some positive integer m such that the base-ten representation of $m!$ ends with exactly n zeroes. How many positive integers less than $1992$ are {\em not} factorial tails?\end{pro} \begin{pro} Prove that if $m$ and $n$ are relatively prime positive integers then $$ \dfrac{(m + n - 1)!}{m!n!}$$is an integer.\end{pro} \begin{pro} If $p$ is a prime divisor of $\binom{2n}{n}$ with $p \geq \sqrt{2n}$ prove that the exponent of p in the factorisation of $\binom{2n}{n}$ equals $1$.\end{pro} \begin{pro} Prove that $$ {\rm lcm} \left( \binom{n}{1}, \binom{n}{2}, \ldots , \binom{n}{n}\right) = \dfrac{{\rm lcm} (1, 2, \ldots , n + 1)}{n + 1}.$$ \end{pro} \begin{pro} Prove the following result of Catalan: $\binom{m + n}{n}$ divides $\binom{2m}{m}\binom{2n}{n}.$ \end{pro} \end{multicols} \section{Complementary Sequences} We define the {\em spectrum} of a real number $\alpha$ to be the infinite multiset of integers $$Spec(\alpha) = \{ \floor{ \alpha }, \floor{ 2\alpha }, \floor{ 3\alpha }, \ldots\} .$$ Two sequences $Spec(\alpha )$ and $Spec(\beta )$ are said to be {\em complementary} if they partition the natural numbers, i.e. $Spec(\alpha )\cap Spec(\beta) = \varnothing$ and $Spec(\alpha ) \cup Spec(\beta ) = \BBN$. \bigskip For example, it appears that the two sequences $$Spec(\sqrt{2}) = \{1, 2, 4, 5, 7, 8, 9, 11, 12, 14, 15, 16, 18, 19, 21, 22, 24, 25, \ldots\} ,$$and $$Spec(2 + \sqrt{2}) = \{3, 6, 10, 13, 17, 20, 23, 27, 30, 34, 37, 40, 44, 47, 51, \ldots\}$$ are complementary. The following theorem establishes a criterion for spectra to be complementary. \begin{thm}[Beatty's Theorem, 1926] If $\alpha > 1$ is irrational and $$\dfrac{1}{\alpha} + \dfrac{1}{\beta} = 1,$$ then the sequences $$ Spec(\alpha ) \ {\rm and} \ Spec(\beta )$$ are complementary. \end{thm} \begin{pf} Since $\alpha > 1, \beta > 1, Spec(\alpha )$ and $Spec(\beta )$ are each sequences of distinct terms, and the total number of terms not exceeding $N$ taken together in both sequences is $\floor{ N/\alpha } + \floor{ N/\beta }$. But $N/\alpha - 1 + N/\beta - 1 < \floor{ N/\alpha }+ [N/\beta ] 1$ (and so $\beta > 1$ also). If $Spec(\alpha ) \cap Spec(\beta )$ is finite, then $$ \lim _{n\rightarrow\infty} \dfrac{\floor{ n/\alpha } + \floor{ n/\beta }}{n} = 1,$$ but since $(\floor{ n/\alpha } + \floor{ n/\beta })\dfrac{1}{n} \rightarrow 1/\alpha + 1/\beta$ as $n \rightarrow \infty$, it follows that $1/\alpha + 1/\beta = 1.$ \end{pf} \begin{exa} Suppose we sieve the positive integers as follows: we choose $a_1 = 1$ and then delete $a_1 + 1 = 2.$ The next term is $3$, which we call $a_2$, and then we delete $a_2 + 2 = 5.$ Thus the next available integer is $4 = a_3$, and we delete $a_3 + 3 = 7$, etc. Thereby we leave the integers $1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, \ldots .$ Find a formula for $a_n$. \end{exa} Solution: What we are asking for is a sequence $\{ S_n\}$ which is complementary to the sequence $\{ S_n + n\}$. By Beatty's Theorem, $\floor{ n\tau }$ and $\floor{ n\tau } + n = \floor{ n(\tau + 1)}$ are complementary if $1/\tau + 1/(\tau + 1) = 1.$ But then $\tau = (1 + \sqrt{5})/2$, the Golden ratio. The $n$-th term is thus $a_n = \floor{ n\tau }.$ \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro}(Skolem) Let $\tau = \dfrac{1 + \sqrt{5}}{2}$ be the Golden Ratio. Prove that the three sequences ($n \geq 1$) $\{ \floor{ \tau \floor{ \tau n }}\} , \{ \floor{ \tau \floor{ \tau ^2 n}}\} , \{ \floor{ \tau ^2 n}\}$ are complementary.\end{pro} \end{multicols} \section{Arithmetic Functions} An {\em arithmetic} function $f$ is a function whose domain is the set of positive integers and whose range is a subset of the complex numbers. The following functions are of considerable importance in Number Theory: \bigskip \begin{tabular}{ll} $d(n)$ & {\em the number of positive divisors of the number n.} \\ $\sigma (n)$ & {\em the sum of the positive divisors of n.} \\ $\phi (n)$ & {\em the number of positive integers not exceeding} \\ & {\em n and relative prime to n.} \\ $\omega (n)$ & {\em the number of distinct prime divisors of n.} \\ $\Omega (n)$ & {\em the number of primes dividing n, counting multiplicity.} \\ \end{tabular} \bigskip In symbols the above functions are: $$ d(n) = \sum _{d|n}1, \ \sigma (n) = \sum _{d|n}d, \ \omega (n) = \sum _{p|n}1, \ \Omega (n) = \sum _{p^\alpha ||n} \alpha, $$ and $$ \phi (n) = \sum _{\stackrel{1 \leq k \leq n}{(k, n) = 1}}1.$$ (The symbol $||$ in $p^\alpha ||n$ is read {\em exactly divides} and it signifies that $p^\alpha |n$ but $p^{\alpha + 1} \not |n.$) \bigskip For example, since $1$, $2$, $4$, $5$, $10$ and $20$ are the divisors of $20$, we have $d(20) = 6$, $\sigma (20) = 42$, $\omega (20) = 2$, $\Omega (20) = 3.$ Since the numbers $1, 3$, $7, 9$, $11, 13$, $17, 19$ are the positive integers not exceeding $20$ and relatively prime to $20$, we see that $\phi (20) = 8.$ \bigskip If $f$ is an arithmetic function which is not identically $0$ such that $f(mn) = f(m)f(n)$ for every pair of relatively prime natural numbers $m, n$, we say that $f$ is then a {\em multiplicative function.} If $f(mn) = f(m)f(n)$ for every pair of natural numbers $m, n$ we say then that $f$ is {\em totally multiplicative}. \bigskip Let $f$ be multiplicative and let $n$ have the prime factorisation $n = p_1 ^{a_1}p_2 ^{a_2}\cdots p_r ^{a_r}$. Then $$f(n) = f(p_1 ^{a_1})f(p_2 ^{a_2}) \cdots f(p_r ^{a_r}).$$ A multiplicative function is thus determined by its values at prime powers. If $f$ is multiplicative, then there is a positive integer $a$ such that $f(a) \neq 0.$ Hence $f(a) = f(1\cdot a) = f(1)f(a)$ which entails that $f(1) = 1.$ \bigskip We will now show that the functions $d$ and $\sigma$ are multiplicative. For this we need first the following result. \bigskip \begin{thm} Let $f$ be a multiplicative function and let $F(n) = \sum _{d|n} f(d).$ Then $F$ is also multiplicative.\end{thm} \begin{pf} Suppose that $a, b$ are natural numbers with $(a, b) = 1$. By the Fundamental Theorem of Arithmetic, every divisor $d$ of $ab$ has the form $d = d_1d_2$ where $d_1|a, d_2|b, (d_1, d_2) =1.$ Thus there is a one-to-one correspondence between positive divisors $d$ of $ab$ and pairs $d_1, d_2$ of positive divisors of $a$ and $b$. Hence, if $n = ab, (a, b) = 1$ then $$ F(n) = \sum _{d|n}f(d) = \sum _{d_1|a} \ \sum _{d_2|b} f(d_1d_2).$$Since $f$ is multiplicative the dextral side of the above equals $$ \sum _{d_1|a}\ \sum _{d_2|b}f(d_1)f(d_2) = \sum _{d_1|a} f(d_1)\sum _{d_2|b}f(d_2) = F(a)F(b).$$ This completes the proof. \end{pf} \bigskip Since the function $f(n) = 1$ for all natural numbers $n$ is clearly multiplicative (indeed, totally multiplicative), the theorem above shows that $d(n) = \sum _{d|n}1$ is a multiplicative function. If $p$ is a prime, the divisors of $p^a$ are $1, p , p^2, p^3, \ldots ,p^a$ and so $d (p^a) = a + 1.$ This entails that if $n$ has the prime factorisation $n = p_1 ^{a_1}p_2 ^{a_2}\cdots p_r ^{a_r}$, then $$d(n) = (1 + a_1)(1 + a_2)\cdots (1 + a_r).$$ For example, $d(2904) = d(2^3\cdot 3\cdot 11^2) = d(2^3)d(3)d(11^2) = (1 + 3)(1 + 1)(1 + 2) = 24.$ \bigskip We give now some examples pertaining to the divisor function. \begin{exa}[AHSME 1993] For how many values of $n$ will an $n$-sided polygon have interior angles with integral degree measures? \end{exa} Solution: The measure of an interior angle of a regular $n$-sided polygon is $\dfrac{(n - 2)180}{n}$. It follows that $n$ must divide $180$. Since there are $18$ divisors of $180$, the answer is $16$, because $n \geq 3$ and so we must exclude the divisors $1$ and $2$. \begin{exa} Prove that $d(n) \leq 2\sqrt{n}$.\end{exa} Solution: Each positive divisor $a$ of $n$ can paired with its complementary divisor $\dfrac{n}{a}$. As $n = a\cdot\dfrac{n}{a}$, one of these divisors must be $\leq \sqrt{n}$. This gives at most $2\sqrt{n}$ divisors. \begin{exa} Find all positive integers n such that $d(n) = 6$. \end{exa} Solution: Since 6 can be factored as $2\cdot 3$ and $6\cdot 1$, the desired $n$ must have only two distinct prime factors, $p$ and $q$, say. Thus $n = p^\alpha q^\beta$ and either $1 + \alpha = 2, 1 + \beta = 3$ or $1 + \alpha = 6, 1 + \beta = 1.$ Hence, $n$ must be of one of the forms $pq^2$ or $p^5,$ where $p, q$ are distinct primes. \begin{exa} Prove that $$\sum _{k = 1} ^n d(k) = \sum _{j = 1} ^n \floor{ \dfrac{n}{j} }$$\end{exa}Solution: We have $$ \sum _{k = 1} ^n { d}(k) = \sum _{k = 1} ^n \sum _{j|k} \ 1.$$Interchanging the order of summation $$ \sum _{j \leq n} \ \sum _{\stackrel{ j \leq k \leq n}{k \equiv 0 \ \mod \ j}} \ 1 = \sum _{j \leq n} \floor{ \dfrac{n}{j} },$$which is what we wanted to prove. \begin{exa}[Putnam 1967] A certain locker room contains $n$ lockers numbered $1, 2, \ldots , n$ and are originally locked. An attendant performs a sequence of operations $T_1, T_2, \ldots , T_n$ whereby with the operation $T_k, 1 \leq k \leq n,$ the condition of being locked or unlocked is changed for all those lockers and only those lockers whose numbers are multiples of $k$. After all the n operations have been performed it is observed that all lockers whose numbers are perfect squares (and only those lockers) are now open or unlocked. Prove this mathematically.\end{exa} Solution: Observe that locker $m, 1 \leq m \leq n$, will be unlocked after $n$ operations if and only if $m$ has an odd number of divisors. Now, $d(m)$ is odd if and only if $m$ is a perfect square. The assertion is proved. \bigskip Since the function $f(n) = n$ is multiplicative (indeed, totally multiplicative), the above theorem entails that $\sigma$ is multiplicative. If $p$ is a prime, then clearly $\sigma (p^a) = 1 + p + p^2 + \cdots + p^a.$ This entails that if $n$ has the prime factorisation $n = p_1 ^{a_1}p_2 ^{a_2} \cdots p_r ^{a_r}$, then $$\sigma (n) = (1 + p_1 + p_1 ^2 + \cdots + p_1 ^{a_1})(1 + p_2 + p_2 ^2 + \cdots + p_w ^{a_2}) \cdots (1 + p_r + p_r ^2 + \cdots + p_r ^{a_r}).$$This last product also equals $$\dfrac{p_1 ^{a_1 + 1} - 1}{p_1 - 1}\cdot\dfrac{p_2 ^{a_2 + 1} - 1}{p_2 - 1}\cdots\dfrac{p_r ^{a_r + 1} - 1}{p_r - 1}. $$ \bigskip We present now some examples related to the function $\sigma$. \begin{exa}[Putnam 1969] Let $n$ be a positive integer such that $24|n + 1.$ Prove that the sum of all divisors of $n$ is also divisible by $24$. \end{exa} Solution: Since $24|n + 1,$ $n \equiv 1$ or $2\mod 3$ and $d \equiv 1, 3, 5$ or $7\mod 8$. As $d(\dfrac{n}{d}) \equiv -1\mod 3$ or $\mod 8$, the only possibilities are $$ \begin{array}{llll} d \equiv 1, & n/d \equiv 2 \ \mod \ 3 & {\rm or \ vice\ versa}, \\ d \equiv 1, & n/d \equiv 7 \ \mod \ 8 & {\rm or\ vice\ versa}, \\ d \equiv 3, & n/d \equiv 5 \ \mod \ 8 & {\rm or\ vice\ versa}.\end{array}$$In all cases $d + n/d \equiv 0\mod 3$ and $\mod 8$, whence $24$ divides $d + n/d.$ As $d\not\equiv n/d,$ no divisor is used twice in the pairing. This implies that $24|\sum _{d|n} d$. \bigskip We say that a natural number is {\em perfect} if it is the sum of its proper divisors. For example, $6$ is perfect because $6 = \sum _{d|6, d\neq 6} d = 1 + 2 + 3.$ It is easy to see that a natural number is perfect if and only if $2n = \sum _{d|n} d.$ The following theorem is classical. \bigskip \begin{thm} An even number is perfect if and only if it is of the form $2^{p - 1}(2^p - 1)$ where both $p$ and $2^p - 1$ are primes. \end{thm} \begin{pf} Suppose that $p, 2^p - 1$ are primes. Then $\sigma (2^p - 1) = 1 + 2^p - 1$. Since $(2^{p - 1}, 2^p - 1) = 1, \sigma (2^{p - 1}(2^p - 1)) = \sigma (2^{p - 1})\sigma (2^p - 1) = (1 + 2 + 2^2 + \cdots + 2^{p - 1})(1 + 2^p - 1) = (2^p - 1)2(2^{p - 1})$, and $2^{p - 1}(2^p - 1)$ is perfect. \bigskip Conversely, let $n$ be an even perfect number. Write $n = 2^sm, m$ odd. Then $\sigma (n) = \sigma (2^s)\sigma (m) = (2^{s + 1} - 1)\sigma (m).$ Also, since $n$ perfect is, $\sigma (n) = 2n = 2^{s + 1}m.$ Hence $(2^{s + 1}- 1)\sigma (m) = 2^{s + 1}m.$ One deduces that $2^{s + 1}|\sigma (m),$ and so $\sigma (m) = 2^{s + 1}b$ for some natural number $b.$ But then $(2^{s + 1} - 1)b = m$, and so $b|m, b \neq m.$ \bigskip We propose to show that $b = 1.$ Observe that $b + m = (2^{s + 1} - 1)b + b = 2^{s + 1}b = \sigma (m).$ If $b\not = 1,$ then there are at least three divisors of $m,$ namely $1, b$ and $m,$ which yields $\sigma (m) \geq 1 + b + m,$ a contradiction. Thus $b = 1,$ and so $m = (2^{s + 1} - 1)b = 2^{s + 1} - 1$ is a prime. This means that $2^{s + 1} - 1$ is a Mersenne prime and hence $s + 1$ must be a prime.\end{pf} \begin{exa} Prove that for every natural number n there exist natural numbers $x$ and $y$ such that $x - y \geq n$ and $\sigma (x^2) = \sigma (y^2).$\end{exa} Solution: Let $s \geq n, (s, 10) = 1.$ We take $x = 5s, y = 4s.$ Then $\sigma (x^2) = \sigma (y^2) = 31\sigma (s^2)$. \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro} Find the numerical values of $d(1024), \sigma (1024), \omega (1024),$ $\Omega (1024)$ and $\phi (1024)$. \end{pro} \begin{pro} Describe all natural numbers $n$ such that $d(n) = 10$.\end{pro} \begin{pro} Prove that $$ d(2^n - 1) \geq d(n).$$\end{pro} \begin{pro} Prove that $d(n) \leq \sqrt{3n}$ with equality if and only if $n = 12.$ \end{pro} \begin{pro} Prove that the following {\em Lambert expansion} holds: $$ \sum _{n = 1} ^\infty d(n) t^n = \sum _{n = 1} ^\infty \dfrac{t^n}{1 - t^n}.$$ \end{pro} \begin{pro} Let $d_1 (n) = d(n), d_k (n) = d(d_{k - 1}(n)), k = 2, 3, \ldots$. Describe $d_k (n)$ for sufficiently large $k$.\end{pro} \begin{pro} Let $m \in \BBN$ be given. Prove that the set $$ {\mathscr A} = \{ n \in \BBN : m|d(n)\}$$contains an infinite arithmetic progression.\end{pro} \begin{pro} Let $n$ be a perfect number. Show that $$ \sum _{d|n} \dfrac{1}{d} = 2.$$ \end{pro} \begin{pro} Prove that $$ \prod _{d|n} d = n^{d(n)/2}.$$\end{pro} \begin{pro} Prove that the power of a prime cannot be a perfect number.\end{pro} \begin{pro}[AIME, 1995] Let $n = 2^{31}3^{19}$. How many positive integer divisors of $n^2$ are less than n but do not divide $n$?\end{pro} \begin{pro} Prove that if $n$ is composite, then $\sigma (n) > n + \sqrt{n}$.\end{pro} \begin{pro} Prove that $\sigma (n) = n + k$, $k > 1$ a fixed natural number has only finitely many solutions.\end{pro} \begin{pro} Characterise all $n$ for which $\sigma (n)$ is odd.\end{pro} \begin{pro} Prove that $p$ is a prime if and only if $\sigma (p) = 1 + p$.\end{pro} \begin{pro} Prove that $$ \dfrac{\sigma (n!)}{n!} \geq 1 + \dfrac{1}{2} + \cdots + \dfrac{1}{n}.$$ \end{pro} \begin{pro} Prove that an odd perfect number must have at least two distinct prime factors.\end{pro} \begin{pro}Prove that in an odd perfect number, only one of its prime factors occurs to an odd power; all the others occur to an even power.\end{pro} \begin{pro}Show that an odd perfect number must contain one prime factor $p$ such that, if the highest power of $p$ occurring in $n$ is $p^a,$ both $p$ and $a$ are congruent to $1$ modulo $4$; all other prime factors must occur to an even power.\end{pro} \begin{pro}Prove that every odd perfect number having three distinct prime factors must have two of its prime factors $3$ and $5$.\end{pro} \begin{pro}Prove that there do not exist odd perfect numbers having exactly three distinct prime factors.\end{pro} \begin{pro} Prove that $$ \sum _{k = 1} ^n \sigma (k) = \sum _{j = 1} ^n j\floor{ \dfrac{n}{j} } . $$ \end{pro} \begin{pro} Find the number of sets of positive integers $\{ a, b, c\}$ such that $a\times b\times c = 462$.\end{pro} \end{multicols} \section{Euler's Function. Reduced Residues} Recall that Euler's $\phi (n)$ function counts the number of positive integers $a \leq n$ that are relatively prime to $n$. We will prove now that $\phi$ is multiplicative. This requires more work than that done for $d$ and $\sigma$. \bigskip First we need the following definitions. \bigskip \begin{df} Let $n > 1$. The $\phi (n)$ integers $1 = a_1 < a_2 < \cdots < a_{\phi (n)} = n - 1$ less than $n$ and relatively prime to $n$ are called the {\em canonical reduced residues} modulo $n$. \end{df} \begin{df} A {\em reduced residue system} modulo $n$, $n > 1$ is a set of $\phi (n)$ incongruent integers modulo $n$ that are relatively prime to $n$.\end{df} For example, the canonical reduced residues $\mod 12$ are $1, 5, 7, 11$ and the set $\{ -11, 5, 19, 23\}$ forms a reduced residue system modulo $12$. We are now ready to prove the main result of this section. \begin{thm} The function $\phi$ is multiplicative.\end{thm} \begin{pf} Let $n$ be a natural number with $n = ab, (a, b) = 1.$ We arrange the $ab$ integers $1, 2, \ldots , ab$ as follows. $${\everymath{\displaystyle}\begin{array}{rrrrrrr} 1 & 2 & 3 & \ldots & k & \ldots & a \\ a + 1 & a + 2 & a + 3 & \ldots & a + k & \ldots & 2a \\ 2a + 1 & 2a + 2 & 2a + 3 & \ldots & 2a + k & \ldots & 3a \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ (b - 1)a + 1 & (b - 1)a + 2 & (b - 1)a + 3 & \ldots & (b - 1)a + k & \ldots & ba \\ \end{array}}$$ Now, an integer $r$ is relatively prime to $m$ if and only if it is relatively prime to $a$ and $b$. We shall determine first the number of integers in the above array that are relatively prime to $a$ and find out how may of them are also relatively prime to $b.$ \bigskip There are $\phi (a)$ integers relatively prime to $a$ in the first row. Now consider the $k$-th column, $1 \leq k \leq a.$ Each integer on this column is of the form $ma + k, 0 \leq m \leq b - 1$. As $k \equiv ma + k\mod a,$ $k$ will have a common factor with $a$ if and only if $ma + k$ does. This means that there are exactly $\phi (a)$ columns of integers that are relatively prime to $a$. We must determine how many of these integers are relatively prime to $b$. \bigskip We claim that no two integers $k, a + k, \ldots , (b -1)a + k$ on the $k$-th column are congruent modulo $b$. For if $ia + k \equiv ja + k\mod b$ then $a(i - j) \equiv 0\mod b$. Since $(a, b) = 1$, we deduce that $i - j \equiv 0\mod b$ thanks to Corollary \ref{cor:reducing_modulus}. Now $i, j \in [0, b - 1]$ which implies that $|i - j| < b.$ This forces $i = j.$ This means that the $b$ integers in any of these $\phi (n)$ columns are, in some order, congruent to the integers $0, 1, \ldots , b - 1.$ But exactly $\phi (b)$ of these are relatively prime to $b.$ This means that exactly $\phi (a)\phi (b)$ integers on the array are relatively prime to $ab$, which is what we wanted to show.\end{pf} If $p$ is a prime and $m$ a natural number, the integers $$p, 2p, 3p, \ldots , p^{m - 1}p$$ are the only positive integers $\leq p^m$ sharing any prime factors with $p^m.$ Thus $\phi (p^m) = p^{m} - p^{m - 1}.$ Since $\phi$ is multiplicative, if $n = p_1 ^{a_1} \cdots p_k ^{a_k}$ is the factorisation of $n$ into distinct primes, then $$ \phi (n) = (p_1 ^{a_1} - p_1 ^{a_1 - 1}) \cdots (p_k ^{a_k} - p_k ^{a_k - 1}).$$For example, $\phi (48) = \phi (2^4\cdot 3) = \phi (2^4)\phi (3) = (2^4 - 2^3)(3 - 1) = 16,$ and $\phi (550) = \phi (2\cdot 5^2 \cdot 11) = \phi (2)\cdot \phi (5^2)\cdot \phi (11) = (2 - 1)(5^2 - 5)(11 - 1) = 1\cdot 20 \cdot 10 = 200$. \begin{exa} Let $n$ be a natural number. How many of the fractions $1/n, 2/n, \ldots , (n - 1)/n, n/n$ are irreducible?\end{exa} Solution: This number is clearly $\sum _{k = 1} ^n \phi (k).$ \begin{exa} Prove that for $n > 1$, $$\sum _{\stackrel{1 \leq a \leq n}{(a, n) = 1}}\,\, a = \dfrac{n\phi (n)}{2}.$$ \end{exa} Solution: Clearly if $1 \leq a \leq n$ and $(a, n) = 1,$ $1 \leq n - a \leq n$ and $(n - a, n) = 1$. Thus $$ S = \sum _{\stackrel{1 \leq a \leq n}{(a, n) = 1}} a = \sum _{\stackrel{1 \leq a \leq n}{(a, n) = 1}} n - a,$$whence $$ 2S = \sum _{\stackrel{1 \leq a \leq n}{(a, n) = 1}}n = n\phi (n).$$ The assertion follows. \begin{thm} Let $n$ be a positive integer. Then $\sum _{d|n} \phi (d) = n.$ \end{thm} \begin{pf} For each divisor $d$ of $n$, let $T_d (n)$ be the set of positive integers $\leq n$ whose gcd with $n$ is $d$. As $d$ varies over the divisors of $n$, the $T_d$ partition the set $\{ 1, 2, \ldots , n\}$ and so $$ \sum _{d|n} \ T_d (n) = n.$$We claim that $T_d (n)$ has $\phi (n/d)$ elements. Note that the elements of $T_d (n)$ are found amongst the integers $d, 2d, \ldots \dfrac{n}{d}d.$ If $k\in T_d (n),$ then $k = ad, 1 \leq a \leq n/d$ and $(k, n) = d$. But then $(\dfrac{k}{d}, \dfrac{n}{d}) = 1$. This implies that $(a, \dfrac{n}{d}) = 1$. Therefore counting the elements of $T_d (n)$ is the same as counting the integers $a$ with $1 \leq a \leq n/d, (a, \dfrac{n}{d}) = 1.$ But there are exactly $\phi (n/d)$ such $a.$ We gather that $$ n = \sum _{d|n} \ \phi (n/d).$$But as $d$ runs through the divisors of $n$, $n/d$ runs through the divisors of $n$ in reverse order, whence $n = \sum _{d|n} \phi (n/d) = \sum _{d|n} \phi (d).$\end{pf} \begin{exa} If $p - 1$ and $p + 1$ are twin primes, and $p > 4$, prove that $3 \phi (p) \leq p.$\end{exa} Solution: Observe that $p > 4$ must be a multiple of $6$, so $$ p = 2^a3^bm, \ ab \geq 1, \ (m, 6) = 1.$$We then have $\phi (p) \leq 2^a3^{b - 1}\phi (m) \leq 2^a 3^{b - 1}m = p/3.$ \begin{exa} Let $n \in \BBN$. Prove that the equation $$ \phi (x) = n!$$is soluble.\end{exa} Solution: We want to solve the equation $\phi (x) = n$ with the constraint that $x$ has precisely the same prime factors as $n$. This restriction implies that $\phi (x)/x = \phi (n)/n$. It follows that $x = n^2/\phi (n).$ \bigskip Let $n = \prod _{p^\alpha ||n} p^\alpha$. Then $x = \prod _{p^\alpha ||n}\dfrac{p^\alpha}{p - 1}.$ The integer $x$ will have the same prime factors as $n$ provided that $\prod _{p|n} (p - 1)|n$. It is clear then that a necessary and sufficient condition for $\phi (x) = n$ to be soluble under the restriction that $x$ has precisely the same prime factors as $n$ is $\prod _{p|n}(p - 1)|n$. If $n = k!,$ this last condition is clearly satisfied. An explicit solution to the problem is thus $x = (k!)^2/\phi (k!).$ \begin{exa} Let $\phi _k (n) = \phi (\phi _{k - 1} (n)), k = 1, 2, \ldots ,$ where $\phi _0 (n) = \phi (n)$. Show that $\forall \, k \in \BBN , \phi _k (n) > 1$ for all sufficiently large $n$.\end{exa} Solution: Let $p_1 ^{a_1}p_2 ^{a_2}\cdots p_r ^{a_r}$ be the prime factorisation of $n$. Clearly $$p_1 ^{a_1/2}p_2 ^{a_2/2}\cdots p_r ^{a_r/2} > 2^{r - 1} \geq \dfrac{1}{2}\dfrac{p_1}{p_1 - 1}\cdots\dfrac{p_r}{p_r - 1}.$$ Hence $$ \phi (n) = \dfrac{p_1 - 1}{p_1} \dfrac{p_2 - 1}{p_2} \cdots\dfrac{p_r - 1}{p_r}p_1 ^{a_1}p_2 ^{a_2}\cdots p_r ^{a_r} \geq \dfrac{1}{2}\dfrac{p_1 ^{a_1}p_2 ^{a_2}\cdots p_r ^{a_r}}{p_1 ^{a_1/2}p_2 ^{a_2/2}\cdots p_r ^{a_r/2}}.$$This last quantity equals $\sqrt{n}/2.$ Therefore $\phi _1 (n) > \dfrac{1}{2}\sqrt{\phi (n)} > \dfrac{1}{2}\sqrt{\dfrac{1}{4}\sqrt{n}} = \dfrac{1}{4}n^{1/4}$. In general we can show that $\phi _k (n) > \dfrac{1}{4}n^{2^{-k - 1}}.$ We conclude that $n \geq 2^{2^{k + 2}}$ implies that $\phi _k (n) > 1.$ \begin{exa} Find infinitely many integers $n$ such that $10|\phi (n).$ \end{exa} Solution: Take $n = 11^k, k = 1, 2, \ldots$. Then $\phi (11^k) = 11^k - 11^{k - 1} = 10\cdot 11^{k - 1}$. \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro} Prove that $$ \phi (n) = n\prod _{p|n}\left(1 - \dfrac{1}{p}\right) . $$\end{pro} \begin{pro} Prove that if $n$ is composite then $\phi (n) \leq n - \sqrt{n}$. When is equality achieved?\end{pro} \begin{pro} {\sc (AIME 1992)} Find the sum of all positive rational numbers that are less than $10$ and have denominator $30$ when written in lowest terms. \end{pro} Answer: 400 \begin{pro} Prove that $\phi (n) \geq n2^{-\omega (n)}$. \end{pro} \begin{pro} Prove that $\phi (n) > \sqrt{n}$ for $n > 6$. \end{pro} \begin{pro}If $\phi (n)|n,$ then $n$ must be of the form $2^a 3^b$ for nonnegative integers $a, b.$ \end{pro} \begin{pro} Prove that if $\phi (n)|n - 1$, then $n$ must be squarefree.\end{pro} \begin{pro}[Mandelbrot 1994] Four hundred people are standing in a circle. You tag one person, then skip $k$ people, then tag another, skip $k$, and so on, continuing until you tag someone for the second time. For how many positive values of $k$ less than $400$ will every person in the circle get tagged at least once?\end{pro} \begin{pro} Prove that if $\phi (n)|n - 1$ and $n$ is composite, then $n$ has at least three distinct prime factors.\end{pro} \begin{pro} Prove that if $\phi (n)|n - 1$ and $n$ is composite, then $n$ has at least four prime factors.\end{pro} \begin{pro} For $n > 1$ let $1 = a_1 < a_2 < \cdots < a_{\phi (n)} = n - 1$ be the positive integers less than $n$ that are relatively prime to $n$. Define the Jacobsthal function $$ g(n) := \max _{1 \leq k \leq \phi (n) - 1} a_{k + 1} - a_k$$ to be the maximum gap between the $a_k$. Prove that $\omega (n) \leq g(n).$ \end{pro} (Hint: Use the Chinese Remainder Theorem). \begin{pro} Prove that a necessary and sufficient condition for $n$ to be a prime is that $$ \sigma (n) + \phi (n) = n d(n).$$\end{pro} \end{multicols} \section{Multiplication in $\BBZ_n$} In section 3.5 we saw that $\BBZ_n$ endowed with the operation of addition $+_n$ becomes a group. We are now going to investigate the multiplicative structure of $\BBZ_n$. How to define multiplication in $\BBZ_n$? If we want to multiply ${\sf a}\cdot _n {\sf b}$ we simply multiply $a\cdot b$ and reduce the result $\mod n$. As an example, let us consider Table \ref{table:multiplication_z6}. To obtain ${\sf 4}\cdot _6 {\sf 2}$ we first multiplied $4\cdot 2 = 8$ and then reduced $\mod 6$ obtaining $8 \equiv 2\mod 6$. The answer is thus ${\sf 4}\cdot _6 {\sf 2} = {\sf 2}$. Another look at the table shows the interesting product ${\sf 3}\cdot _6 {\sf 2} = {\sf 0}$. Why is it interesting? We have multiplied to non-zero entities and obtained a zero entity! Does $\BBZ_6$ form a group under $\cdot _6$? What is the multiplicative identity? In analogy with the rational numbers, we would like ${\sf 1}$ to be the multiplicative identity. We would then define the multiplicative inverse of ${\bf a}$ to be that ${\bf b}$ that has the property that ${\sf a}\cdot _6 {\sf b} = {\sf b}\cdot _6 {\sf a} = {\sf 1}.$ But then, we encounter some problems. For example, we see that ${\sf 0}, {\sf 2}, {\sf 3},$ and {\sf 4} do not have a multiplicative inverse. We need to be able to identify the invertible elements of $\BBZ_n$. For that we need the following. \begin{table}[h] \centering \begin{tabular}{|c|c|c|c|c|c|c|} \hline $\cdot _6$ & ${\sf{0}}$ & ${\sf{1}}$ & ${\sf{2}}$ & ${\sf 3}$ & ${\sf 4}$ & ${\sf 5}$ \\ \hline ${\sf{0}}$ & ${\sf{0}}$ & ${\sf{0}}$ & ${\sf{0}}$ & ${\sf 0}$ & ${\sf 0}$ & ${\sf 0}$ \\ \hline ${\sf{1}}$ & ${\sf{0}}$ & ${\sf{1}}$ & ${\sf{2}}$ & ${\sf 3}$ & ${\sf 4}$ & ${\sf 5}$ \\ \hline ${\sf{2}}$ & ${\sf{0}}$ & ${\sf{2}}$ & ${\sf{4}}$ & ${\sf 0}$ & ${\sf 2}$ & ${\sf 4}$ \\ \hline ${\sf{3}}$ & ${\sf{0}}$ & ${\sf{3}}$ & ${\sf{0}}$ & ${\sf 3}$ & ${\sf 0}$ & ${\sf 3}$ \\ \hline ${\sf{4}}$ & ${\sf{0}}$ & ${\sf{4}}$ & ${\sf{2}}$ & ${\sf 0}$ & ${\sf 4}$ & ${\sf 2}$ \\ \hline ${\sf{5}}$ & ${\sf{0}}$ & ${\sf{5}}$ & ${\sf{4}}$ & ${\sf 3}$ & ${\sf 2}$ & ${\sf 1}$ \\ \hline \end{tabular} \vspace{1cm}\footnotesize \hangcaption{Multiplication Table for $\BBZ_6$} \label{table:multiplication_z6} \end{table} \begin{df} Let $n > 1$ be a natural number. An integer $b$ is said to be the inverse of an integer $a$ modulo $n$ if $ab \equiv 1 \mod n$. \end{df} It is easy to see that inverses are unique $\mod n$. For if $x, y$ are inverses to $a\mod n$ then $ax \equiv 1\mod n$ and $ay \equiv 1\mod n$. Multiplying by $y$ the first of these congruences, $(ya)x \equiv y\mod n$. Hence $x \equiv y\mod n.$ \begin{thm} Let $n > 1, a$ be integers. Then $a$ possesses an inverse modulo $n$ if and only if $a$ is relatively prime to $n$. \end{thm} \begin{pf} Assume that $b$ is the inverse of $a\mod n$. Then $ab \equiv 1\mod n$, which entails the existence of an integer $s$ such that $ab - 1 = sn$, i.e. $ab - sn = 1.$ This is a linear combination of $a$ and $n$ and hence divisible by $(a, n)$. This implies that $(a, n) = 1.$ Conversely if $(a, n) = 1$, by the Bachet-Bezout Theorem there are integers $x, y$ such that $ax + ny = 1$. This immediately yields $ax \equiv 1\mod n$, i.e., $a$ has an inverse $\mod n.$\end{pf} \begin{exa} Find the inverse of $5 \mod 7$.\end{exa} Solution: We are looking for a solution to the congruence $5x \equiv 1\mod 7$. By inspection we see that this is $x \equiv 3 \mod 7$. According to the preceding theorem, $a$ will have a multiplicative inverse if and only if $(a, n) = 1.$ We thus see that only the reduced residues $\mod n$ have an inverse. We let $\BBZ_n ^\times = \{{\sf a}_1, {\sf a}_2, \ldots , {\sf a}_{\phi (n)}\}$. It is easy to see that the operation $\cdot _n$ is associative, since it inherits associativity from the integers. We conclude that $\BBZ_n ^\times$ is a group under the operation $\cdot _n$. We now give some assorted examples. \begin{exa}[IMO 1964] Prove that there is no positive integer $n$ for which $2^n + 1$ is divisible by $7$. \end{exa} Solution: Observe that $2^1 \equiv 2, 2^2 \equiv 4$, $2^3 \equiv 1\mod 7$, $2^4 \equiv 2\mod 7$, $2^5 \equiv 4\mod 7$, $2^6 \equiv 1\mod 7$, etc. The pattern $2$, $4$, $1$, repeats thus cyclically. This says that there is no power of $2$ which is $\equiv - 1 \equiv 6\mod 7$. \begin{thm}\label{thm:existence_of_order} If $a$ is relatively prime to the positive integer $n$, there exists a positive integer $k \leq n$ such that $a^k \equiv 1\mod n$. \end{thm} \begin{pf} Since $(a, n) = 1$ we must have $(a^j, n) = 1$ for all $j \geq 1.$ Consider the sequence $a, a^2, a^3, \ldots , a^{n + 1}\mod n$. As there are $n + 1$ numbers and only $n$ residues mod $n,$ the Pigeonhole Principle two of these powers must have the same remainder $\mod n$. That is, we can find $s, t$ with $1 \leq s < t \leq n + 1$ such that $a^s \equiv a^t\mod n$. Now, $1 \leq t - s \leq n.$ Hence $a^s \equiv a^t\mod n$ gives $a^{t - s}a^s \equiv a^{t - s}a^t\mod n$, which is to say $a^t \equiv a^{t - s}a^t\mod n$. Using Corollary \ref{cor:reducing_modulus} we gather that $a^{t - s} \equiv 1\mod n$, which proves the result.\end{pf} \bigskip If $(a, n) = 1$, the preceding theorem tells us that there is a positive integer $k$ with $a^k \equiv 1\mod n$. By the Well-Ordering Principle, there must be a smallest positive integer with this property. This prompts the following definition. \begin{df} If $m$ is the least positive integer with the property that $a^m \equiv 1 \mod n$, we say that $a$ has order $m\mod n$. \end{df} For example, $3^1 \equiv 3, 3^2 \equiv 2, 3^3 \equiv 6, 3^4 \equiv 4, 3^5 \equiv 5, 3^6 \equiv 1\mod 7$, and so the order of $3\mod 7$ is $6$. We write this fact as ord$_7 3 = 6.$ \bigskip Given $n$, not all integers $a$ are going to have an order $\mod n$. This is clear if $n|a,$ because then $a^m \equiv 0\mod n$ for all positive integers $m$. The question as to which integers are going to have an order $\mod n$ is answered in the following theorem. \begin{thm} Let $n > 1$ be a positive integer. Then $a\in \BBZ$ has an order $\mod n$ if and only if $(a, n) = 1$. \end{thm} \begin{pf} If $(a, n) = 1$, then $a$ has an order in view of Theorem \ref{thm:existence_of_order} and the Well-Ordering Principle. Hence assume that $a$ has an order $\mod n$. Clearly $a \neq 0.$ The existence of an order entails the existence of a positive integer $m$ such that $a^m \equiv 1\mod n$. Hence, there is an integer $s$ with $a^m + sn = 1$ or $a\cdot a^{m - 1} + sn = 1$. This is a linear combination of $a$ and $n$ and hence divisible by $(a, n)$. This entails that $(a, n) = 1.$ \end{pf} The following theorem is of utmost importance. \begin{thm} \label{thm:order_divides_powers} Let $(a, n) = 1$ and let $t$ be an integer. Then $a^t \equiv 1 \mod n$ if and only if {\rm ord}$_n a|t$. \end{thm} \begin{pf} Assume that ${\rm ord}_n a|t.$ Then there is an integer $s$ such that $s{\rm ord}_n a = t$. This gives $$a^{t} \equiv a^{s{\rm ord}_n a} \equiv (a^{{\rm ord}_n a})^s \equiv 1^s \equiv 1 \ \mod \ n.$$ \bigskip Conversely, assume that $a^t \equiv 1\mod n$ and $t = x\cdot{\rm ord}_n a + y, 0 \leq y < {\rm ord}_n a. $ Then $$a^y \equiv a^{t - x{\rm ord}_n a} \equiv a^t\cdot (a^{{\rm ord}_n a}) ^{-x} \equiv 1\cdot 1^{-x} \equiv 1 \ \mod \ n.$$ \bigskip If $y > 0$ we would have a positive integer smaller than ${\rm ord}_n a$ with the property $a^y \equiv 1\mod n$. This contradicts the definition of ${\rm ord}_n a$ as the smallest positive integer with that property. Hence $y = 0$ and thus $t = x\cdot{\rm ord}_n a$, i.e., ${\rm ord}_n a|t.$\end{pf} \begin{exa}[IMO 1964] Find all positive integers $n$ for which $2^n - 1$ is divisible by $7$. \end{exa} Solution: Observe that the order of $2\mod 7$ is $3$. We want $2^n \equiv 1\mod 7$. It must then be the case that $3|n$. Thus $n = 3, 6, 9, 12, \ldots$. The following result will be used repeatedly. \begin{thm}\label{thm:reduced_residues} Let $n > 1, a \in \BBZ, (a, n) = 1.$ If $r_1, r_2, \ldots , r_{\phi (n)}$ is a reduced set of residues modulo $n$, then $ar_1, ar_2, \ldots , ar_{\phi (n)}$ is also a reduced set of residues modulo $n$. \end{thm} \begin{pf} We just need to show that the $\phi (n)$ numbers $ar_1, ar_2, \ldots , ar_{\phi (n)}$ are mutually incongruent $\mod n$. Suppose that $ar_i \equiv ar_j\mod n$ for some $i \neq j.$ Since $(a, n) = 1,$ we deduce from Corollary \ref{cor:reducing_modulus} that $r_i \equiv r_j\mod n$. This contradicts the fact that the $r$'s are incongruent, so the theorem follows.\end{pf} For example, as $1, 5, 7, 11$ is a reduced residue system modulo $12$ and $(12, 5) = 1$, the set $5, 25, 35, 55$ is also a reduced residue system modulo $12$. Again, the $1, 5, 7, 11$ are the $5, 25, 35, 55$ in some order and $1\cdot 5\cdot 7\cdot 11 \equiv 5\cdot 25\cdot 35\cdot 55\mod 12$. The following corollary to Theorem \ref{thm:reduced_residues} should be immediate. \begin{cor} \label{cor:reduced_residues} Let $n > 1, a, b \in \BBZ, (a, n) = 1.$ If $r_1, r_2, \ldots , r_{\phi (n)}$ is a reduced set of residues modulo $n$, then $ar_1 + b, ar_2 + b, \ldots , ar_{\phi (n)} + b$ is also a reduced set of residues modulo $n$. \end{cor} \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro} Find the order of $5$ modulo $12$.\end{pro} \end{multicols} \section{M\"{o}bius Function} \begin{df} The {\em M\"{o}bius function} is defined for positive integer n as follows: $$ \mu (n) = \left\{ \begin{array}{ll} 1 & {\rm if} \ n = 1, \\ (-1)^{\omega (n)} & {\rm if} \ \omega (n) = \Omega (n), \\ 0 & {\rm if} \ \omega (n) < \Omega (n). \end{array} \right. $$ \end{df} Thus $\mu$ is 1 for $n = 1$ and square free integers with an even number of prime factors, $-1$ for square free integers with an odd number of prime factors, and 0 for non-square free integers. Thus for example $\mu (6) = 1, \mu (30) = -1$ and $\mu (18) = 0$. \begin{thm}\label{thm:mobius_multiplies} The M\"{o}bius Function $\mu$ is multiplicative. \end{thm} \begin{pf} Assume $(m, n) = 1.$ If both ${\mathscr M}$ and $n$ are square-free then $$ \mu (m)\mu (n) = (-1)^{\omega (m)}(-1)^{\omega (n)} = (-1)^{\omega (m) + \omega (n)} = \mu (mn).$$If one of $m, n$ is not square-free then $$ \mu (m)\mu (n) = 0 = \mu (mn).$$This proves the theorem. \end{pf} \begin{thm}\label{thm:mobius_sifts} $$ \sum _{d|n} \mu (d) = \left\{ \begin{array}{ll} 1 & {\rm if} \ n = 1, \\ 0 & {\rm if} \ n > 1. \end{array} \right. $$ \end{thm} \begin{pf} There are $\binom{\omega (n)}{k}$ square-free divisors $d$ of $n$ with exactly $k$ prime factors. For all such $d, \mu (d) = (-1)^k$. The sum in question is thus $$\sum _{d|n} \mu (d) = \sum _{k = 0} ^{\omega (n)} \binom{\omega (n)}{k}(-1)^k.$$ By the Binomial Theorem this last sum is $(1 - 1)^{\omega (n)} = 0.$\end{pf} \begin{thm}[M\"{o}bius Inversion Formula] \label{thm:mobius_inverts} Let $f$ be an arithmetical function and $F(n) = \sum _{d|n} f(d).$ Then $$ f(n) = \sum _{d|n} \mu (d)F(n/d) = \sum _{d|n} \mu (n/d)F(d).$$\end{thm} \begin{pf} We have $$\begin{array}{lcl}\sum _{d|n} \mu (d)F(n/d) & = & \sum _{d|n} \sum _{d|n} \sum _{s|\dfrac{n}{d}} f(s) \\ & = & \sum _{ds|n} \mu (d) f(s) \\ & = & \sum _{s|n} f(s) \sum _{d|\dfrac{n}{s}} \mu (d) . \end{array}$$ In view of theorem \ref{thm:mobius_sifts}, the inner sum is different from $0$ only when $\dfrac{n}{s} = 1.$ Hence only the term $s = n$ in the outer sum survives, which means that the above sums simplify to $f(n).$\end{pf} \bigskip We now show the converse to Theorem \ref{thm:mobius_inverts}. \bigskip \begin{thm} Let $f$, $F$ be arithmetic functions with $f(n) = \sum _{d|n} \mu (d)F(n/d)$ for all natural numbers $n$. Then $F(n) = \sum _{d|n} f(d)$.\end{thm} \begin{pf} We have $$\begin{array}{lcl}\sum _{d|n} f(d) & = & \sum _{d|n}\sum _{s|d} \mu (s) F(d/s) \\ & = & \sum _{d|n} \sum _{s|d} \mu (d/s) F(s) \\ & = & \sum _{s|n} \sum _{r|\dfrac{n}{s}} \mu (r) F(s). \end{array}$$Using Theorem \ref{thm:mobius_sifts}, the inner sum will be $0$ unless $s = n,$ in which case the entire sum reduces to $F(n).$\end{pf} \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro}Prove that $$ \phi (n) = n\sum _{d|n} \dfrac{\mu (d)}{d}.$$ \end{pro} \begin{pro} If f is an arithmetical function and $F(n) = \sum _{k = 1} ^n f([n/k]),$ then $$ f(n) = \sum _{j = 1} ^{n} \mu (j) F([n/j]) .$$ \end{pro} \begin{pro} If F is an arithmetical function such that $f(n) = \sum _{k = 1} ^n \mu (k)F([n/k]),$ prove that $F(n) = \sum _{j = 1} ^n f(j)$.\end{pro} \begin{pro} Prove that $\sum _{d|n}|\mu (d)| = 2^{\omega (n)}.$ \end{pro} \begin{pro} Prove that $\sum _{d|n} \mu (d)d(d) = (-1)^{\omega (n)}$.\end{pro} \begin{pro} Given any positive integer k, prove that there exist infinitely many integers n with $$ \mu (n + 1) = \mu (n + 2) = \cdots = \mu (n + k).$$\end{pro} \end{multicols} \chapter{More on Congruences} \section{Theorems of Fermat and Wilson} \begin{thm}[Fermat's Little Theorem] Let $p$ be a prime and let $p\not |a$. Then $$ a^{p - 1} \equiv 1 \ \mod \ p. $$ \end{thm} \begin{pf} Since $(a, p) = 1$, the set $a\cdot 1, a\cdot 2, \ldots , a\cdot (p - 1)$ is also a reduced set of residues $\mod p$ in view of Theorem \ref{thm:reduced_residues}. Hence $$ (a\cdot 1)(a\cdot 2)\cdots (a\cdot (p - 1)) \equiv 1\cdot 2 \cdots (p - 1) \ \mod \ p,$$or $$ a^{p - 1}(p - 1)! \equiv (p - 1)! \ \mod \ p.$$ As $((p - 1)!, p) = 1$ we may cancel out the $(p - 1)!$'s in view of Corollary \ref{cor:reducing_modulus}. This proves the theorem.\end{pf} As an obvious corollary, we obtain the following. \begin{cor} For every prime $p$ and for every integer a, $$a^p \equiv a \ \mod \ p. $$ \end{cor} \begin{pf} Either $p|a$ or $p\not |a$. If $p|a, a \equiv 0 \equiv a^p\mod p$ and there is nothing to prove. If $p\not|a$, Fermat's Little Theorem yields $p|a^{p - 1} - 1$. Hence $p|a(a^{p - 1} - 1) = a^p - a,$ which again gives the result.\end{pf} The following corollary will also be useful. \begin{cor}\label{cor:order_divides_p-1} Let $p$ be a prime and $a$ an integer. Assume that $p\not|a$. Then ${\rm ord}_p a|p - 1.$ \end{cor} \begin{pf} This follows immediately from Theorem \ref{thm:order_divides_powers} and Fermat's Little Theorem.\end{pf} \begin{exa} Find the order of $8 \mod 11$.\end{exa} Solution: By Corollary \ref{cor:order_divides_p-1} ${\rm ord}_{11} 8$ is either $1, 2, 5$ or 10. Now $8^2 \equiv -2\mod 11, 8^4 \equiv 4\mod 11$ and $8^5 \equiv -1\mod 11$. The order is thus ${\rm ord}_{11} 8 = 10.$ \begin{exa} Let $a_1 = 4, a_{n} = 4^{a_{n - 1}}, n > 1.$ Find the remainder when $a_{100}$ is divided by $7$. \end{exa} Solution: By Fermat's Little Theorem, $4^6 \equiv 1\mod 7$. Now, $4^n \equiv 4\mod 6$ for all positive integers $n$, i.e., $4^n = 4 + 6t$ for some integer $t$. Thus $$ a_{100} \equiv 4^{a_{99}} \equiv 4^{4 + 6t} \equiv 4^4\cdot (4^6)^t \equiv 4 \ \mod \ 7.$$ \begin{exa} Prove that for $m, n \in \BBZ$, $mn(m^{60} - n^{60})$ is always divisible by $56786730$. \end{exa} Solution: Let $a = 56786730$ = $2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13\cdot 31\cdot 61$. Let $Q(x, y) = xy(x^{60} - y^{60})$. Observe that $(x - y)|Q(x, y)$, $(x^2 - y^2)|Q(x, y)$, $(x^3 - y^3)|Q(x, y)$, $(x^4 - y^4)|Q(x, y)$, $(x^6 - y^6)|Q(x, y)$, $(x^{10} - y^{10})|Q(x, y)$, $(x^{12} - y^{12})|Q(x, y)$, and $(x^{30} - y^{30})|Q(x, y)$. If $p$ is any one of the primes dividing $a$, the Corollary to Fermat's Little Theorem yields $m^{p} - m \equiv 0\mod p$ and $n^{p} - n \equiv 0\mod p$. Thus $n(m^{p} - m) - m(n^{p} - n) \equiv 0\mod p$, i.e., $mn(m^{p - 1} - n^{p - 1}) \equiv 0\mod p.$ Hence, we have $2|mn(m - n)|Q(m, n), 3|mn(m^2 - n^2)|Q(m,n), 5|mn(m^4 - n^4)|Q(m, n), 7|mn(m^6 - n^6)|Q(m, n), 11|mn(m^{10} - n^{10})|Q(m, n), 13|mn(m^{12} - n^{12})|Q(m, n), 31|mn(m^{30} - n^{30})|Q(m, n)$ and $61|mn(m^{60} - n^{60})|Q(m, n).$ Since these are all distinct primes, we gather that $a|mnQ(m, n)$, which is what we wanted. \begin{exa}[Putnam 1972] Show that given an odd prime $p$, there are always infinitely many integers $n$ for which $p|n2^n + 1.$ \end{exa} Answer: For any odd prime $p$, take $n = (p - 1)^{2k + 1}, k = 0, 1, 2, \ldots$. Then $$n2^n + 1 \equiv (p - 1)^{2k + 1}(2^{p - 1})^{(p - 1)^{2k}} + 1 \equiv (-1)^{2k + 1}1^{2k} + 1 \equiv 0 \ \mod \ p.$$ \begin{exa} Prove that there are no integers $n > 1$ with $n|2^n - 1.$ \end{exa} Solution: If $n|2^n - 1$ for some $n > 1,$ then $n$ must be odd and have a smallest odd prime $p$ as a divisor. By Fermat's Little Theorem, $2^{p - 1} \equiv 1\mod p$. By Corollary \ref{cor:order_divides_p-1} , ord$_p 2$ has a prime factor in common with $p - 1.$ Now, $p|n|2^n - 1$ and so $2^n \equiv 1\mod p.$ Again, by Corollary \ref{cor:order_divides_p-1}, ord$_p 2$ must have a common prime factor with $n$ (clearly ord$_p 2 > 1$). This means that $n$ has a smaller prime factor than $p$, a contradiction. \begin{exa}Let $p$ be a prime. Prove that \begin{enumerate} \item $$ \binom{p - 1}{n} \equiv (-1)^n \ \mod \ p, \ 1 \leq n \leq p - 1.$$ \item $$ \binom{p + 1}{n} \equiv 0 \ \mod \ p, \ 2 \leq n \leq p - 1.$$ \item If $p \neq 5$ is an odd prime, prove that either $f_{p - 1}$ or $f_{p + 1}$ is divisible by p.\end{enumerate}\end{exa} Solution: (1) $(p - 1)(p - 2)\cdots (p - n) \equiv (-1)(-2) \cdots (-n) \equiv (-1)^n n!\mod p$. The assertion follows from this. \\ (2) $(p + 1)(p)(p - 1) \cdots (p - n + 2) \equiv (1)(0)(-1) \cdots (-n + 2) \equiv 0\mod p$. The assertion follows from this. \\ (3) Using the Binomial Theorem and Binet's Formula $$ f_n = \dfrac{1}{2^{n - 1}}\left( \binom{n}{1} + 5\binom{n}{3} + 5^2\binom{n}{5} + \cdots\right) .$$ From this and (1), $$ 2^{p - 2}f_{p - 1} \equiv p - 1 - (5 + 5^2 + \cdots + 5^{(p - 3)/2}) \equiv -\dfrac{5^{(p - 1)/2} - 1}{4} \ \mod \ p.$$ Using (2),$$2^pf_{p + 1} \equiv p + 1 + 5^{(p - 1)/2} \equiv 5^{(p - 1)/2} + 1 \ \mod \ p.$$ Thus $$ 2^pf_{p - 1}f_{p + 1} \equiv 5^{p - 1} - 1 \ \mod \ p. $$ But by Fermat's Little Theorem, $5^{p - 1} \equiv 1\mod p$ for $p \neq 5.$ The assertion follows. \begin{lem} If $a^2 \equiv 1 \mod p$, then either $a \equiv 1 \mod p$ or $a \equiv -1 \mod p$.\end{lem} \begin{pf} We have $p|a^2 - 1 = (a - 1)(a + 1).$ Since $p$ is a prime, it must divide at least one of the factors. This proves the lemma.\end{pf} \begin{thm}[Wilson's Theorem]\label{thm:wilson} If $p$ is a prime, then $(p - 1)! \equiv -1 \mod p$. \end{thm} \begin{pf} If $p = 2$ or $p = 3,$ the result follows by direct verification. So assume that $p > 3.$ Consider $a, 2 \leq a \leq p - 2.$ To each such $a$ we associate its unique inverse $\overline{a}\mod p$, i.e. $a\overline{a} \equiv 1\mod p$. Observe that $a \neq \overline{a}$ since then we would have $a^2 \equiv 1\mod p$ which violates the preceding lemma as $a \neq 1, a \neq p - 1$. Thus in multiplying all $a$ in the range $2 \leq a \leq p - 2$, we pair them of with their inverses, and the net contribution of this product is therefore $1$. In symbols, $$ 2 \cdot 3 \cdots (p - 2) \equiv 1 \ \mod \ p.$$ In other words, $$ (p - 1)! \equiv 1\cdot \left(\prod _{ 2 \leq a \leq p - 2}j\right)\cdot (p - 1) \equiv 1\cdot 1\cdot (p - 1) \equiv -1 \ \mod \ p.$$ This gives the result. \end{pf} \begin{exa} If $p \equiv 1 \mod 4$, prove that $$ \left(\dfrac{p - 1}{2}\right) ! \equiv - 1 \ \mod \ p.$$\end{exa} Solution: In the product $(p - 1)!$ we pair off $j, 1 \leq j \leq (p - 1)/2$ with $p - j$. Observe that $j(p - j) \equiv -j^2\mod p$. Hence $$ -1 \equiv (p - 1)! \equiv \prod _{1 \leq j \leq (p - 1)/2} -j^2 \equiv (-1)^{(p - 1)/2}\left(\dfrac{p - 1}{2}\right) ! \ \mod \ p.$$ As $(-1)^{(p - 1)/2} = 1$, we obtain the result. \begin{exa}[IMO 1970] Find the set of all positive integers $n$ with the property that the set $$\{ n, n + 1, n + 2, n + 3, n + 4, n + 5\} $$ can be partitioned into two sets such that the product of the numbers in one set equals the product of the numbers in the other set. \end{exa} Solution: We will show that no such partition exists. Suppose that we can have such a partition, with one of the subsets having product of its members equal to $A$ and the other having product of its members equal to $B$. We might have two possibilities. The first possibility is that exactly one of the numbers in the set $\{ n, n + 1, n + 2, n + 3, n + 4, n + 5\}$ is divisible by $7$, in which case exactly one of $A$ or $B$ is divisible by $7$, and so $A\cdot B$ is not divisible by $7^2$, and so $A\cdot B$ is not a square. The second possibility is that all of the members of the set are relatively prime to 7. In this last case we have $$ n(n + 1)\cdots (n + 6) \equiv 1\cdot 2\cdots 6 \equiv A\cdot B\equiv -1 \ \mod \ 7.$$ But if $A = B$ then we are saying that there is an integer $A$ such that $A^2 \equiv -1\mod 7$, which is an impossibility, as $-1$ is not a square $\mod 7$. This finishes the proof. \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro} Find all the natural numbers $n$ for which $3|(n2^n + 1)$. \end{pro} \begin{pro} Prove that there are infinitely many integers $n$ with $n|2^n + 2.$ \end{pro} \begin{pro} Find all primes $p$ such that $p|2^p + 1.$ \end{pro} Answer: $p = 3$ only. \begin{pro} If $p$ and $q$ are distinct primes prove that $$pq|(a^{pq} - a^p - a^q - a)$$ for all integers $a$. \end{pro} \begin{pro} If $p$ is a prime prove that $p|a^p + (p - 1)!a$ for all integers $a$.\end{pro} \begin{pro} If $(mn, 42) = 1$ prove that $168|m^6 - n^6.$\end{pro} \begin{pro} Let $p$ and $q$ be distinct primes. Prove that $$ q^{p - 1} + p^{q - 1} \equiv 1 \ \mod \ pq.$$\end{pro} \begin{pro} If $p$ is an odd prime prove that $n^p \equiv n \mod 2p$ for all integers $n$.\end{pro} \begin{pro} If $p$ is an odd prime and $p|m^p + n^p$ prove that $p^2|m^p + n^p.$\end{pro} \begin{pro} Prove that $n > 1$ is a prime if and only if $(n - 1)! \equiv -1 \mod n$.\end{pro} \begin{pro} Prove that if $p$ is an odd prime $$ 1^2\cdot 3^2 \cdots (p - 2)^2 \equiv 2^2\cdot4^2 \cdots (p - 1)^2 \equiv (-1)^{(p - 1)/2}\ \mod \ p$$\end{pro} \begin{pro} Prove that $19|(2^{2^{6k + 2}} + 3)$ for all nonnegative integers $k$.\end{pro} \end{multicols} \section{Euler's Theorem} In this section we obtain a generalisation of Fermat's Little Theorem, due to Euler. The proof is analogous to that of Fermat's Little Theorem. \begin{thm}[Euler's Theorem]\label{thm:euler} Let $(a, n) = 1$. Then $a^{\phi (n)} \equiv 1 \mod n$.\end{thm} \begin{pf} Let $a_1, a_2, \ldots , a_{\phi (n)}$ be the canonical reduced residues $\mod n$. As $(a, n) = 1$, $aa_1, aa_2, \ldots , aa_{\phi (n)}$ also forms a set of incongruent reduced residues. Thus $$aa_1\cdot aa_2\cdots aa_{\phi (n)} \equiv a_1a_2\cdots a_{\phi (n)} \ \mod \ n,$$ or $$ a^{\phi (n)}a_1a_2\cdots a_{\phi (n)} \equiv a_1a_2\cdots a_{\phi (n)} \ \mod n.$$As $(a_1a_2\cdots a_{\phi (n)}, n) = 1$, we may cancel the product $a_1a_2\cdots a_{\phi (n)}$ from both sides of the congruence to obtain Euler's Theorem.\end{pf} Using Theorem \ref{thm:euler} we obtain the following corollary. \begin{cor} \label{cor:order_divides_phi}Let $(a, n) = 1$. Then ${\rm ord}_n a|\phi (n).$\end{cor} \begin{exa}Find the last two digits of $3^{1000}.$ \end{exa} Solution: As $\phi (100) = 40$, by Euler's Theorem, $3^{40} \equiv 1\mod 100$. Thus $$ 3^{1000} = (3^{40})^{25} \equiv 1^{25} = 1 \ \mod \ 100,$$and so the last two digits are $01$. \begin{exa}Find the last two digits of $7^{7^{1000}}$.\end{exa} Solution: First observe that $\phi (100) = \phi (2^2)\phi (5^2) = (2^2 - 2)(5^2 - 5) = 40.$ Hence, by Euler's Theorem, $7^{40} \equiv 1\mod 100$. Now, $\phi (40) = \phi (2^3)\phi (5) = 4\cdot 4 = 16$, hence $7^{16} \equiv 1\mod 40$. Finally, $1000 = 16\cdot 62 + 8.$ This means that $7^{1000} \equiv (7^{16})^{62}7^8 \equiv 1^{62}7^8 \equiv (7^4)^2 \equiv 1^2 \equiv 1\mod 40$. This means that $7^{1000} = 1 + 40t$ for some integer $t.$ Upon assembling all this $$ 7^{7^{1000}} \equiv 7^{1 + 40t} \equiv 7\cdot (7^{40})^t \equiv 7 \ \mod \ 100.$$ This means that the last two digits are $07$. \begin{exa}[IMO 1978] $m, n$ are natural numbers with $1 \leq m < n.$ In their decimal representations, the last three digits of $1978^m$ are equal, respectively, to the last three digits of $1978^n$. Find $m, n$ such that $m + n$ has its least value. \end{exa} Solution: As $m + n = n - m + 2m$, we minimise $n - m.$ We are given that $$ 1978^n - 1978^m = 1978^m(1978^{n - m} - 1)$$is divisible by $1000 = 2^35^3.$ Since the second factor is odd, $2^3$ must divide the first and so $m \geq 3.$ Now, ord$_{125} 1978$ is the smallest positive integer $s$ with $$1978^s \equiv 1 \ \mod \ 125.$$ By Euler's Theorem $$ 1978^{100} \equiv 1 \ \mod \ 125$$ and so by Corollary 7.3 $s|100.$ Since $125|(1978^s - 1)$ we have $5|(1978^s - 1)$, i.e., $1978^s \equiv 3^s \equiv 1\mod 5$. Since $s|100$, this last congruence implies that $s = 4, 20,$ or $100$. We now rule out the first two possibilities. \bigskip Observe that $$ 1978^4 \equiv (-22)^4 \equiv 2^4\cdot 11^4 \equiv (4\cdot 121)^2 \equiv (-16)^2 \equiv 6 \ \mod \ 125.$$ This means that $s \neq 4.$ Similarly $$1978^{20} \equiv 1978^4\cdot (1978^{4})^4 \equiv 6\cdot 6^4 \equiv 6\cdot 46 \equiv 26 \ \mod \ 125.$$ This means that $s \neq 20$ and so $s = 100$. Since $s$ is the smallest positive integer with $1978^s \equiv 1\mod 125$, we take $n - m = s = 100$ and $m = 3$, i.e., $n = 103, m = 3$, and finally, $m + n = 106.$ \begin{exa}[IMO 1984] Find one pair of positive integers $a, b$ such that:\\ (i) $ab(a + b)$ is not divisible by $7$. \\ (ii) $(a + b)^7 - a^7 - b^7$ is divisible by $7^7$. Justify your answer. \end{exa} Solution: We first factorise $(a + b)^7 - a^7 - b^7$ as $ab(a + b)(a^2 + ab + b^2)^2$. Using the Binomial Theorem we have $$\begin{array}{lcl} (a + b)^7 - a^7 - b^7 & = & 7(a^6b + ab^6 + 3(a^5b^2 + a^2b^5) + 5(a^4b^3 + a^3b^4)) \\ & = & 7ab(a^5 + b^5 + 3ab(a^3 + b^3) + 5(a^2b^2)(a + b)) \\ & = & 7ab(a + b)(a^4 + b^4 - a^3b - ab^3 + a^2b^2 \\ & & \qquad + 3ab(a^2 - ab + b^2) + 5ab) \\ & = & 7ab(a + b)(a^4 + b^4 + 2(a^3b + ab^3) + 3a^2b^2) \\ & = & 7ab(a + b)(a^2 + ab + b^2)^2. \end{array}$$ The given hypotheses can be thus simplified to $${\rm (i)'} \ ab(a + b) \ {\rm is\ not\ divisible\ by\ } 7,$$ $${\rm (ii)'} \ a^2 + ab + b^2 \ {\rm is\ divisible\ by\ } 7^3.$$ As $(a + b)^2 > a^2 + ab + b^2 \geq 7^3,$ we obtain $a + b \geq 19.$ Using trial and error, we find that $a = 1, b = 18$ give an answer, as $1^2 + 1\cdot 18 + 18^2 = 343 = 7^3.$ \bigskip Let us look for more solutions by means of Euler's Theorem. As $a^3 - b^3 = (a - b)(a^2 + ab + b^2),$ (ii)' is implied by $${\rm (ii)''} \ \left\{\begin{array}{c} a^3 \equiv b^3 \ \mod \ 7^3 \\ a \not\equiv b \ \mod \ 7. \end{array}\right.$$ Now $\phi (7^3) = (7 - 1)7^2 = 3\cdot 98,$ and so if $x$ is not divisible by $7$ we have $(x^{98})^3 \equiv 1\mod 7^3,$ which gives the first part of (ii)'. We must verify now the conditions of non-divisibility. For example, letting $x = 2$ we see that $2^{98} \equiv 4\mod 7$. Thus letting $a = 2^{98}, b = 1$. Letting $x = 3$ we find that $3^{98} \equiv 324\mod 7^3$. We leave to the reader to verify that $a = 324, b = 1$ is another solution. \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro} Show that for all natural numbers $s$, there is an integer $n$ divisible by $s$, such that the sum of the digits of $n$ equals $s$. \end{pro} \begin{pro} Prove that $504|n^9 - n^3.$ \end{pro} \begin{pro} Prove that for odd integer $n > 0,$ $n|(2^{n!} - 1)$. \end{pro} \begin{pro} Let $p\not|10$ be a prime. Prove that $p$ divides infinitely many numbers of the form $$ 11\ldots 11.$$ \end{pro} \begin{pro}Find all natural numbers $n$ that divide $$ 1^n + 2^n + \cdots + (n - 1)^n.$$\end{pro} \begin{pro} Let $(m, n) = 1$. Prove that $$ m^{\phi (n)} + n^{\phi (n)} \equiv 1 \ \mod \ mn.$$\end{pro} \begin{pro}Find the last two digits of $a_{1001}$ if $a_1 = 7, a_n = 7^{a_{n - 1}}.$\end{pro} \begin{pro} Find the remainder of $$ 10^{10} + 10^{10^2} + \cdots + 10^{10^{10}}$$ upon division by $7$.\end{pro} \begin{pro} Prove that for every natural number $n$ there exists some power of $2$ whose final $n$ digits are all ones and twos. \end{pro} \begin{pro}[USAMO 1982] Prove that there exists a positive integer $k$ such that $k\cdot 2^n + 1$ is composite for every positive integer $n$.\end{pro} \begin{pro}[Putnam 1985] Describe the sequence $a_1 = 3 , a_n = 3^{a_{n - 1}}\mod 100$ for large $n$.\end{pro} \end{multicols} \chapter{Scales of Notation} \section{The Decimal Scale} As we all know, any natural number $n$ can be written in the form $$ n = a_010^k + a_110^{k - 1} + \cdots + a_{k - 1}10 + a_k,$$where $1 \leq a_0 \leq 9, 0 \leq a_j \leq 9, j \geq 1.$ For example, $65789 = 6\cdot 10^4 + 5\cdot 10^3 + 7\cdot 10^2 + 8\cdot 10 + 9.$ \begin{exa} Find all whole numbers which begin with the digit $6$ and decrease $25$ times when this digit is deleted. \end{exa} Solution: Let the number sought have $n + 1$ digits. Then this number can be written as $6\cdot 10^n + y,$ where $y$ is a number with $n$ digits (it may begin with one or several zeroes). The condition of the problem stipulates that $$ 6\cdot 10^n + y = 25\cdot y$$whence $$ y = \dfrac{6\cdot 10^n}{24}.$$ From this we gather that $n \geq 2$ (otherwise, $6\cdot 10^n$ would not be divisible by 24). For $n \geq 2, y = 25\cdot 10^{k - 2}$, that is, $y$ has the form $250\cdots 0 (n - 2$ zeroes). We conclude that all the numbers sought have the form $625\underbrace{0\ldots 0}_{n-2 \ {\rm zeroes}}$. \begin{exa}[IMO 1968] Find all natural numbers $x$ such that the product of their digits (in decimal notation) equals $x^2 - 10x - 22.$ \end{exa} Solution: Let $x$ have the form $$ x = a_0 + a_1 10 + a_2 10^2 + \cdots + a_{n - 1}10^{n - 1}, \,\, \,\,\, a_k \leq 9, a_{n - 1} \neq 0.$$ Let $P(x)$ be the product of the digits of $x$, $P(x) = x^2 - 10x - 22.$ Now, $P(x) = a_0 a_1 \cdots a_{n - 1} \leq 9^{n - 1}a_{n - 1} < 10^{n - 1}a_{n - 1} \leq x$ (strict inequality occurs when $x$ has more than one digit). So $x^2 - 10x - 22 < x,$ and we deduce that $x < 13,$ whence $x$ has either one digit or $x = 10, 11, 13$. If $x$ had one digit, then $a_0 = x^2 - 10x - 22,$ but this equation has no integral solutions. If $x = 10, P(x) = 0, $ but $x^2 - 10x - 22 \neq 0.$ If $x = 11, P(x) = 1, $ but $x^2 - 10x - 22 \neq 1.$ If $x = 12, P(x) = 2$ and $x^2 - 10x - 22 = 2$. Therefore, $x = 12$ is the only solution. \begin{exa} A whole number decreases an integral number of times when its last digit is deleted. Find all such numbers. \end{exa} Solution: Let $0 \leq y \leq 9,$ and $10x + y = mx, m$ and $x$ natural numbers. This requires $10 + y/x = m,$ an integer. We must have $x|y.$ If $y = 0$, any natural number $x$ will do, and we obtain the multiples of 10. If $y = 1, x = 1,$ and we obtain $11$. If $y = 2, x = 1$ or $x = 2$ and we obtain $12$ and $22$. Continuing in this fashion, the sought numbers are: the multiples of $10, 11$, $12, 13$, $14, 15$, $16, 17$, $18, 19$,$22, 24$, $26, 28$, $33, 36$, $39, 44$, $48, 55$, $66, 77, 88$, and $99$. \begin{exa} Let $A$ be a positive integer, and $A'$ be a number written with the aid of the same digits with are arranged in some other order. Prove that if $A + A' = 10^{10}$, then $A$ is divisible by $10$. \end{exa} Solution: Clearly $A$ and $A'$ must have ten digits. Let $A = a_{10}a_9\ldots a_1$ be the consecutive digits of $A$ and $A' = a_{10}'a' _9 \ldots a_1 '.$ Now, $A + A' = 10^{10}$ if and only if there is a $j, 0 \leq j \leq 9$ for which $a_1 + a_1' = a_2 + a_2' = \cdots = a_j + a_j' = 0, a_{j + 1} + a_{j + 1}' = 10, a_{j + 2} + a_{j + 2}' = a_{j + 3} + a_{j + 3}' = \cdots = a_{10} + a_{10}' = 9.$ Notice that $j = 0$ implies that there are no sums of the form $a_{j + k} + a_{j + k}', k \geq 2,$ and $j = 9$ implies that there are no sums of the form $a_l + a_l', 1\leq l \leq j.$ On adding all these sums, we gather $$ a_1 + a_1' + a_2 + a_2' + \cdots + a_{10} + a_{10}' = 10 + 9(9 - j).$$Since the $a_s'$ are a permutation of the $a_s$, we see that the sinistral side of the above equality is the even number $2(a_1 + a_2 + \cdots + a_{10}).$ This implies that $j$ must be odd. But this implies that $a_1 + a_1' = 0$, which gives the result. \begin{exa}[AIME 1994] Given a positive integer $n$, let $p(n)$ be the product of the non-zero digits of $n$. (If $n$ has only one digit, then $p(n)$ is equal to that digit.) Let $$ S = p(1) + p(2) + \cdots + p(999).$$ What is the largest prime factor of $S$? \end{exa} Solution: Observe that {\em non-zero} digits are the ones that matter. So, for example, the numbers 180, 108, 118, 810, 800, and 811 have the same value $p(n).$ We obtain all the three digit numbers from $001$ to $999$ by expanding the product $$ (0 + 1 + 2 + \cdots + 9)^3 - 0,$$where we subtracted a $0$ in order to eliminate $000$. Thus $$ (0 + 1 + 2 \cdots + 9)^3 - 0 = 001 + 002 + \cdots + 999.$$In order to obtain $p(n)$ for a particular number, we just have to substitute the (possible) zeroes in the decimal representation, by 1's, and so $$ p(1) + p(2) + \cdots + p(n) = 111 + 112 + \cdots + 999 = (1 + 1 + 2 + \cdots + 9)^3 - 1,$$ which equals $46^3 - 1.$ (In the last sum, 111 is repeated various times, once for 001, once for 011, once for 100, once for 101, once for 110, etc.) As $46^3 - 1 = 3^3\cdot 5\cdot 7\cdot 103,$ the number required is $103$. \begin{exa}[AIME 1992] Let $S$ be the set of all rational numbers $r, 0 < r < 1,$ that have a repeating decimal expansion of the form $$ 0.abcabcabc\ldots = 0.\overline{abc}, $$ where the digits $a, b, c$ are not necessarily distinct. To write the elements of $S$ as fractions in lowest terms, how many different numerators are required? \end{exa} Solution: Observe that $0.abcabcabc\ldots = \dfrac{abc}{999}$, and $999 = 3^3\cdot 37.$ If $abc$ is neither divisible by 3 nor 37, the fraction is already in lowest terms. By the Inclusion-Exclusion Principle, there are $$ 999 - (999/3 + 999/37) + 999/3\cdot 37 = 648$$such numbers. Also, fractions of the form $s/37,$ where $3|s, 37\not |s$ are in $S$. There are $12$ fractions of this kind. (Observe that we do not consider fractions of the form $l/3^t, 37|s, 3\not |l,$ because fractions of this form are greater than $1$, and thus not in $S$.) The total number of distinct numerators in the set of reduced fractions is thus $640 + 12 = 660$. \begin{exa}[Putnam 1956] Prove that every positive integer has a multiple whose decimal representation involves all $10$ digits. \end{exa} Solution: Let $n$ be an arbitrary positive integer with $k$ digits. Let $m = 123456789\cdot 10^{k + 1}$. Then all of the $n$ consecutive integers $m + 1, m + 2, \ldots m + n$ begin with 1234567890 and one of them is divisible by $n$. \begin{exa}[Putnam 1987] The sequence of digits $$ 12345678910111213141516171819202122 \ldots $$ is obtained by writing the positive integers in order. If the $10^n$ digit of this sequence occurs in the part in which the $m$-digit numbers are placed, define $f(n)$ to be $m$. For example $f(2) = 2$, because the hundredth digit enters the sequence in the placement of the two-digit integer $55$. Find, with proof, $f(1987)$. \end{exa} Solution: There are $9\cdot 10^{j - 1} j$-digit positive integers. The total number of digits in numbers with at most $r$ digits is $g(r) = \sum _{j = 1} ^{r} j\cdot 9\cdot 10^{r - 1} = r10^r - \dfrac{10^r - 1}{9}$. As $0 < \dfrac{10^r - 1}{9} < 10^r$, we get $(r - 1)10^r < g(r) < r10^r.$ Thus $g(1983) < 1983\cdot 10^{1983} < 10^4 \cdot 10^{1983} = 10^{1987}$ and $g(1984) > 1983\cdot 10^{1984} > 10^3\cdot 10^{1984}.$ Therefore $f(1987) = 1984.$ \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro} Prove that there is no whole number which decreases $35$ times when its initial digit is deleted. \end{pro} \begin{pro} A whole number is equal to the arithmetic mean of all the numbers obtained from the given number with the aid of all possible permutations of its digits. Find all whole numbers with that property. \end{pro} \begin{pro}[AIME 1989] Suppose that $n$ is a positive integer and $d$ is a single digit in base-ten. Find $n$ if $$ \dfrac{n}{810} = 0.d25d25d25d25\ldots .$$\end{pro} \begin{pro}[AIME 1992] For how many pairs of consecutive integers in $$\{ 1000, 1001, \ldots , 2000\}$$ is no carrying required when the two integers are added? \end{pro} \begin{pro} Let $m$ be a seventeen-digit positive integer and let $N$ be number obtained from $m$ by writing the same digits in reversed order. Prove that at least one digit in the decimal representation of the number $M + N$ is even. \end{pro} \begin{pro}Given that $$ e = 2 + \dfrac{1}{2!} + \dfrac{1}{3!} + \dfrac{1}{4!} + \cdots ,$$ prove that $e$ is irrational. \end{pro} \begin{pro} Let $t$ be a positive real number. Prove that there is a positive integer $n$ such that the decimal expansion of $nt$ contains a $7$. \end{pro} \begin{pro} [AIME 1988] Find the smallest positive integer whose cube ends in $888$. \end{pro} \begin{pro}[AIME 1987] An ordered pair $(m, n)$ of nonnegative integers is called {\em simple} if the addition $m + n$ requires no carrying. Find the number of simple ordered pairs of nonnegative integers that sum $1492$. \end{pro} \begin{pro}[AIME 1986] In the parlor game, the ``magician'' asks one of the participants to think of a three-digit number $abc$, where $a, b, c$ represent the digits of the number in the order indicated. The magician asks his victim to form the numbers $acb, bac, cab$ and $cba$, to add the number and to reveal their sum $N$. If told the value of $N$, the magician can identity $abc$. Play the magician and determine $abc$ if $N = 319$. \end{pro} \begin{pro} The integer $n$ is the smallest multiple of $15$ such that every digit of $n$ is either $0$ or $8$. Compute $n/15$. \end{pro} \begin{pro}[AIME 1988] For any positive integer $k$, let $f_1 (k)$ denote the square of the sums of the digits of $k$. For $n \geq 2,$ let $f_n (k) = f_1 (f_{n - 1}(k))$. Find $f_{1988}(11)$. \end{pro} \begin{pro}[IMO 1969] Determine all three-digit numbers N that are divisible by $11$ and such that $N/11$ equals the sum of the squares of the digits of $N.$\end{pro} \begin{pro}[IMO 1962] Find the smallest natural number having last digit is $6$ and if this $6$ is erased and put in front of the other digits, the resulting number is four times as large as the original number. \end{pro} \begin{pro}\begin{enumerate}\item Show that {\em Champernowne's number} $$ \chi = 0.123456789101112131415161718192021 \ldots $$ is irrational. \item Let $r \in \BBQ$ and let $\epsilon > 0$ be given. Prove that there exists a positive integer $n$ such that $$ |10^n \chi - r| < \epsilon .$$ \end{enumerate} \end{pro} \begin{pro} A {\em Liouville number} is a real number $x$ such that for every positive $k$ there exist integers $a$ and $b \geq 2,$ such that $$ |x - a/b| < b^{-k}.$$ Prove or disprove that $\pi$ is the sum of two Liouville numbers.\end{pro} \begin{pro} Given that $$ 1/49 = 0.020408163265306122448979591836734693877551,$$ find the last thousand digits of $$ 1 + 50 + 50^2 + \cdots + 50^{999}.$$\end{pro} \end{multicols} \section{Non-decimal Scales} The fact that most people have ten fingers has fixed our scale of notation to the decimal. Given any positive integer $r > 1$, we can, however, express any number in base $r$. \begin{exa} Express the decimal number $5213$ in base-seven. \end{exa} Solution: Observe that $5213 < 7^5$. We thus want to find $0 \leq a_0 , \ldots , a_4 \leq 6, a_4 \neq 0,$ such that $$ 5213 = a_4 7^4 + a_3 7^3 + a_2 7^2 + a_1 7 + a_0 .$$ Now, divide by $7^4$ to obtain $$ 2 + {\rm proper\,\, fraction} = a_4 + {\rm proper\,\, fraction} .$$ Since $a_4$ is an integer, it must be the case that $a_4 = 2$. Thus $5213 - 2\cdot 7^4 = 411 = a_3 7^3 + a_2 7^2 + a_1 7 + a_0$. Dividing 411 by $7^3$ we obtain $$ 1 + {\rm proper\,\, fraction} = a_3 + {\rm proper\,\, fraction}.$$ Thus $a_3 = 1.$ Continuing in this way we deduce that $5213 = 21125_{7}$. \begin{exa} Express the decimal number $13/16$ in base-six. \end{exa} Solution: Write $$ \dfrac{13}{16} = \dfrac{a_1}{6} + \dfrac{a_2}{6^2} + \dfrac{a_3}{6^3} + \ldots .$$ Multiply by 6 to obtain $$ 4 + {\rm proper\,\, fraction} = a_1 + {\rm proper\,\, fraction}.$$ Thus $a_1 = 4.$ Hence $13/16 - 4/6 = 7/48 = \dfrac{a_2}{6^2} + \dfrac{a_3}{6^3} + \ldots$. Multiply by $6^2$ to obtain $$ 5 + {\rm proper\,\, fraction} = a_2 + {\rm proper\,\, fraction}.$$ We gather that $a_2 = 5.$ Continuing in this fashion, we deduce that $13/16 = .4513_{6}.$ \begin{exa} Prove that $4.41$ is a perfect square in any scale of notation. \end{exa} Solution: If 4.41 is in scale $r$, then $$ 4.41 = 4 + \dfrac{4}{r} + \dfrac{1}{r^2} = \left( 2 + \dfrac{1}{r}\right)^2 .$$ \begin{exa} Let $\floor{ x } $ denote the greatest integer less than or equal to $x$. Does the equation $$\floor{ x } + \floor{ 2x } + \floor{ 4x } + \floor{ 8x } + \floor{ 16x } + \floor{ 32x } = 12345 $$ have a solution? \end{exa} Solution: We show that there is no such $x$. Recall that $\floor{ x }$ satisfies the inequalities $x - 1 < \floor{ x } \leq x.$ Thus \begin{eqnarray*}x - 1 + 2x - 1 + 4x - 1 + \cdots + 32x - 1 & < & \floor{ x } + \floor{ 2x } + \floor{ 4x } + \floor{ 8x } \\ & & \qquad + \floor{ 16x } + \floor{ 32x } \\ & \leq & x + 2x + 4x + \cdots + 32x .\end{eqnarray*} From this we see that $63x - 6 < 12345 \leq 63x.$ Hence $195 < x < 196$. Write then $x$ in base-two: $$ x = 195 + \dfrac{a_1}{2} + \dfrac{a_2}{2^2} + \dfrac{a_3}{2^3} + \ldots ,$$ with $a_k = 0$ or 1. Then $$\begin{array}{lll} \floor{ 2x } & = & 2\cdot 195 + a_1 , \\ \floor{ 4x } & = & 4\cdot 195 + 2a_1 + a_2 , \\ \floor{ 8x } & = & 8\cdot 195 + 4a_1 + 2a_2 + a_3 , \\ \floor{ 16x } & = & 16\cdot 195 + 8a_1 + 4a_2 + 2a_3 + a_4 , \\ \floor{ 32x } & = & 32\cdot 195 + 16a_1 + 8a_2 + 4a_3 + 2a_4 + a_5 . \end{array}$$ Adding we find that $\floor{ x } + \floor{ 2x } + \floor{ 4x } + \floor{ 8x } + \floor{ 16x } + \floor{ 32x } = 63\cdot 195 + 31a_1 + 15a_2 + 7a_3 + 3a_4 + a_5 ,$ i.e. $31a_1 + 15a_2 + 7a_3 + 3a_4 + a_5 = 60$. This cannot be because $31a_1 + 15a_2 + 7a_3 + 3a_4 + a_5 \leq 31 + 15 + 7 + 3 + 1 = 57 < 60.$ \begin{exa}[AHSME 1993] Given $0 \leq x_0 < 1,$ let $$ x_n = \left\{ \begin{array}{ll} 2x_{n - 1} & {\rm if} \,\, 2x_{n - 1} < 1 \\ 2x_{n - 1} - 1 & {\rm if}\,\, 2x_{n - 1} \geq 1 \end{array}\right.$$ for all integers $n > 0.$ For how many $x_0$ is it true that $x_0 = x_5$? \end{exa} Solution: Write $x_0$ in base-two, $$ x_0 = \sum _{k = 1} ^\infty \dfrac{a_n}{2^n} \,\, a_n = 0 \,\, {\rm or}\,\, 1.$$ The algorithm given just moves the binary point one unit to the right. For $x_0$ to equal $x_5$ we need $0.a_1 a_2 a_3 a_4 a_5a_6a_7 \ldots = 0.a_6a_7a_8a_9a_{10}a_{11}a_{12}\ldots$. This will happen if and only if $x_0$ has a repeating expansion with $a_1 a_2 a_3 a_4 a_5$ as the repeating block . There are $2^5 = 32$ such blocks. But if $a_1 = a_2 = \cdots = a_5 = 1,$ then $x_0 = 1,$ which is outside $[0, 1)$. The total number of values for which $x_0 = x_5$ is thus $32 - 1 = 31.$ \begin{exa}[AIME 1986] The increasing sequence $$1, 3, 4, 9, 10, 12, 13, \ldots$$ consists of all those positive integers which are powers of $3$ or sums distinct powers of $3$. Find the hundredth term of the sequence. \end{exa} Solution: If the terms of the sequence are written in base-$3$, they comprise the positive integers which do not contain the digit $2$. Thus, the terms of the sequence in ascending order are thus $$ 1, 10, 11, 100, 101, 110, 111, \ldots .$$ In the {\em binary} scale, these numbers are, of course, 1, 2, 3, \ldots . To obtain the $100$-th term of the sequence we just write $100$ in binary $100 = 1100100_2$ and translate this into ternary: $1100100_3 = 3^6 + 3^5 + 3^2 = 981.$ \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro}[Putnam, 1987] For each positive integer $n$, let $\alpha (n)$ be the number of zeroes in the base-three representation of $n$. For which positive real numbers $x$ does the series $$ \sum _{n = 1} ^\infty \dfrac{x^{\alpha (n)}}{n^3}$$ converge? \end{pro} \begin{pro} Prove that for $x \in \BBR , x \geq 0,$ one has $$ \sum _{n = 1} ^\infty \dfrac{(-1)^{\floor{2^n x}}}{2^n} = 1 - 2(x - \floor{x}).$$ \end{pro} \begin{pro}[Putnam, 1981] Let $E(n)$ denote the largest $k$ such that $5^k$ is an integral divisor of $1^1 2^2 3^3 \cdots n^n$. Calculate $$\lim _{n\rightarrow \infty} \dfrac{E(n)}{n^2}.$$ \end{pro} \begin{pro}[AHSME, 1982] The base-eight representation of a perfect square is $ab3c$ with $a \neq 0$. Find the value of $c$. \end{pro} \begin{pro}[Putnam, 1977] An ordered triple of $(x_1, x_2 , x_3)$ of positive irrational numbers with $x_1 + x_2 + x_3 = 1$ is called balanced if $x_n < 1/2$ for all $1 \leq n \leq 3$. If a triple is not balanced, say $x_j > 1/2$, one performs the following ``balancing act'': $$ B(x_1 , x_2 , x_3 ) = (x_1 ', x_2 ', x_3 '),$$ where $x_i ' = 2x_i$ if $x_i \neq x_j$ and $x_j ' = 2x_j - 1$. If the new triple is not balanced, one performs the balancing act on it. Does continuation of this process always lead to a balanced triple after a finite number of performances of the balancing act? \end{pro} \begin{pro} Let $C$ denote the class of positive integers which, when written in base-three, do not require the digit $2$. Show that no three integers in $C$ are in arithmetic progression. \end{pro} \begin{pro} Let $B(n)$ be the number of $1$'s in the base-two expansion of $n$. For example, $B(6) = B(110_2 ) = 2, B(15) = B(1111_2 ) = 4$. \begin{enumerate} \item {\sc (Putnam 1981)} Is $$ \exp \left( \sum _{n = 1} ^\infty \dfrac{B(n)}{n^2 + n}\right)$$ a rational number? \item {\sc (Putnam 1984)} Express $$ \sum _{n = 0} ^{2^m - 1} \, (-1)^{B(n)} n^m $$ in the form $(-1)^m a^{f(m)}(g(m))!$ where $a$ is an integer and $f, g$ are polynomials. \end{enumerate} \end{pro} \begin{pro}What is the largest integer that I should be permitted to choose so that you may determine my number in twenty ``yes'' or ``no'' questions?\end{pro} \end{multicols} \section{A theorem of Kummer} We first establish the following theorem. \begin{thm}[Legendre] \label{thm:legendre} Let $p$ be a prime and let $n = a_0p^k + a_1p^{k - 1} + \cdots + a_{k - 1}p + a_k$ be the base-$p$ expansion of $n$. The exact power m of a prime p dividing $n!$ is given by $$ m =\dfrac{n - (a_0 + a_1 + \cdots + a_k)}{p - 1}.$$\end{thm} \begin{pf} By De Polignac's Formula $$ m = \sum _{k = 1} ^\infty \floor{ \dfrac{n}{p^k} }.$$Now, $\floor{n/p} = a_0p^{k - 1} + a_1p^{k - 2} + \cdots a_{k - 2}p + a_{k - 1}, \floor{n/p^2} = a_0p^{k - 2} + a_1p^{k - 3} + \cdots + a_{k - 2}, \ldots , \floor{n/p^k} = a_0.$ Thus $$ {\everymath{\displaystyle}\begin{array}{lcl}\sum _{k = 1} ^\infty \floor{n/p^k} & = & a_0 (1 + p + p^2 + \cdots + p^{k - 1}) + a_1(1 + p + p^2 + \cdots + p^{k - 2}) + \\ & & \quad \cdots + a_{k - 1}(1 + p) + a_{k} \\ & = & a_0\dfrac{p^k - 1}{p - 1} + a_1\dfrac{p^{k - 1} - 1}{p - 1} + \cdots + a_{k - 1}\dfrac{p^2 - 1}{p - 1} + a_k\dfrac{p - 1}{p - 1} \\ & = & \dfrac{a_0p^k + a_1p^{k - 1} + \cdots + a_k - (a_0 + a_1 + \cdots + a_k)}{p - 1} \\ & = & \dfrac{n - (a_0 + a_1 + \cdots + a_k)}{p - 1}, \end{array}}$$as wanted.\end{pf} \begin{thm}[Kummer's Theorem]\label{thm:kummer} The exact power of a prime $p$ dividing the binomial coefficient $\binom{a + b}{a}$ is equal to the number of ``carry-overs'' when performing the addition of $a, b$ written in base $p$. \end{thm} \begin{pf} Let $a = a_0 + a_1p + \cdots + a_kp^k, b = b_0 + b_1p + \cdots + b_kp^k, 0 \leq a_j, b_j \leq p - 1,$ and $a_k+ b_k > 0.$ Let $S_a = \sum _{j = 0} ^k a_j, S_b = \sum _{j = 0} ^k b_j.$ Let $c_j, 0 \leq c_j \leq p - 1,$ and $\epsilon _j = 0$ or 1, be defined as follows: $$ \begin{array}{l} a_0 + b_0 = \epsilon _0p + c_0, \\ \epsilon _0 + a_1 + b_1 = \epsilon _1p + c_1, \\ \epsilon _1 + a_2 + b_2 = \epsilon _2p + c_2, \\ \vdots \\ \epsilon _{k - 1} + a_k + b_k = \epsilon _kp + c_k. \\ \end{array}$$ Multiplying all these equalities successively by $1, p, p^2, \ldots$ and adding them: $$ \begin{array}{lll}a + b + \epsilon _0p + \epsilon _1p^2 + \ldots + \epsilon _{k - 1}p^k & = & \epsilon _0p + \epsilon _1p^2 + \ldots + \epsilon _{k - 1}p^k + \epsilon _k p^{k + 1} \\ & & + c_0 + c_1p + \cdots + c_kp^k \end{array}.$$ We deduce that $a + b = c_0 + c_1p + \cdots + c_kp^k + \epsilon _kp^{k + 1}.$ By adding all the equalities above, we obtain similarly: $$ S_a + S_b + (\epsilon _0 + \epsilon _1 + \cdots + \epsilon _{k - 1}) = (\epsilon _0 + \epsilon _1 + \cdots + \epsilon _{k})p + S_{a + b} - \epsilon _k.$$ Upon using Legendre's result from above, $$(p - 1)m = (a + b) - S_{a + b} - a + S_a - b + S_b = (p - 1)(\epsilon _0 + \epsilon _1 + \cdots + \epsilon _{k}),$$ which gives the result.\end{pf} \chapter{Miscellaneous Problems} \begin{exa} Prove that $$ \sum _{\stackrel{p}{p \ {\rm prime} }} \dfrac{1}{p}$$diverges.\end{exa} Solution: Let ${\mathscr F}_x$ denote the family consisting of the integer 1 and the positive integers $n$ all whose prime factors are less than or equal to $x$. By the Unique Factorisation Theorem \begin{equation} \prod _{\stackrel{p \leq x}{p \ {\rm prime}}} \left( 1 + \dfrac{1}{p} + \dfrac{1}{p^2} + \cdots \right) = \sum _{ n \in {\mathscr F}_x} \dfrac{1}{n}.\end{equation}Now, $$ \sum _{n \in {\mathscr F}_x} \dfrac{1}{n} > \sum _{n \leq x} \dfrac{1}{n}. $$As the harmonic series diverges, the product on the sinistral side of 2.3.3 diverges as $x \rightarrow \infty$. But $$ \prod _{\stackrel{p \leq x}{p \ {\rm prime}}} \left( 1 + \dfrac{1}{p} + \dfrac{1}{p^2} + \cdots \right) = \sum _{\stackrel{p \leq x}{p \ {\rm prime}}} \dfrac{1}{p} + O(1).$$This finishes the proof. \begin{exa} Prove that for each positive integer $k$ there exist infinitely many even positive integers which can be written in more than $k$ ways as the sum of two odd primes. \end{exa} Solution: Let $a_k$ denote the number of ways in which $2k$ can be written as the sum of two odd primes. Assume that $a_k \leq C \ \forall \, k$ for some positive constant $C$. Then $$ \left( \sum _{\stackrel{\scriptstyle p > 2}{p \ {\rm prime}}} x^p \right) ^2 = \sum _{k = 2} ^\infty a_k x^{2k} \leq C \dfrac{x^4}{1 - x^2}.$$This yields $$ \sum _{\stackrel{\scriptstyle p > 2}{p \ {\rm prime}}} x^{p - 1} \leq \sqrt{C} \dfrac{x}{\sqrt{1 - x^2}}.$$Integrating term by term, $$ \sum _{\stackrel{\scriptstyle p > 2}{p \ {\rm prime}}} \dfrac{1}{p} \leq \sqrt{C} \int _0 ^1 \dfrac{x}{\sqrt{1 - x^2}} \, dx = \sqrt{C}.$$ But the leftmost series is divergent, and we obtain a contradiction. \begin{exa}[IMO 1976] Determine, with proof, the largest number which is the product of positive integers whose sum is $1976$.\end{exa} Solution: Suppose that $$ a_1 + a_2 + \cdots + a_n = 1976;$$we want to maximise $ \displaystyle{\prod _{k = 1} ^n a_k}$. We shall replace some of the $a_k$ so that the product is enlarged, but the sum remains the same. By the arithmetic mean-geometric mean inequality $$ \left(\prod _{k = 1} ^n a_k \right)^{1/n} \leq \dfrac{a_1 + a_2 + \cdots + a_n }{n},$$ with equality if and only if $a_1 = a_2 = \cdots = a_n$. Thus we want to make the $a_k$ as equal as possible. \bigskip If we have an $a_k \geq 4,$ we replace it by two numbers $2, a_k - 2$. Then the sum is not affected, but $2(a_k - 2) \geq a_k ,$ since we are assuming $a_k \geq 4$. Therefore, in order to maximise the product, we must take $a_k = 2$ or $a_k = 3$. We must take as many $2$'s and $3$'s as possible. \bigskip Now, $2 + 2 + 2 = 3 + 3,$ but $2^3 < 3^2$, thus we should take no more than two 2's. Since $1976 = 3\cdot 658 + 2,$ the largest possible product is $2\cdot 3^{658}$. \begin{exa}[USAMO 1983] Consider an {\em open} interval of length $1/n$ on the real line, where $n$ is a positive integer. Prove that the number of irreducible fractions $a/b, 1 \leq b \leq n,$ contained in the given interval is at most $(n + 1)/2.$\end{exa} Solution: Divide the rational numbers in $(x, x + 1/n)$ into two sets: $\{\dfrac{s_k}{t_k}\}, k = 1, 2, \ldots , r,$ with denominators $1 \leq t_k \leq n/2$ and those $u_k /v_k , k = 1, 2 , \ldots , s$ with denominators $n /2 < v_k \leq n$, where all these fractions are in reduced form. Now, for every $t_k$ there are integers $c_k$ such that $n/2 \leq c_k t_k \leq n$. Define $u_{s + k} = c_k s_k , v_{s + k} = c_k t_k , y_{k + r} = u_{k + r}/v_{k + r}$. No two of the $y_l , 1 \leq l \leq r + s$ are equal, for otherwise $y_j = y_k$ would yield $$ |u_k /v_k - u_i /v_i | \geq 1/v_i \geq 1/n,$$ which contradicts that the open interval is of length $1/n$. Hence the number of distinct rationals is $r + s \leq n - \floor{n/2} \leq (n + 1)/2$. \bigskip {\em Aliter:} Suppose to the contrary that we have at least $\floor{(n + 1)/2} + 1 = a$ fractions. Let $s_k , t_k , 1 \leq k \leq a$ be the set of numerators and denominators. The set of\ denominators is a subset of $$\{ 1, 2, \ldots , 2(a - 1)\}. $$ By the Pigeonhole Principle, $t_i |t_k$ for some $i, k,$ say $t_k = mt_i$. But then $$ |s_k /t_k - s_i /t_i| = |ms_i - s_k|/t_k \geq 1/n, $$contradicting the hypothesis that the open interval is of length $1/n$. \begin{exa} Let $$ Q_{r, s} = \dfrac{(rs)!}{r!s!}.$$Show that $Q_{r, ps} \equiv Q_{r, s} \,\, \mod \, p$, where $p$ is a prime \end{exa} Solution: As $$ Q_{r, s} = \prod _{j = 1} ^r \binom{js - 1}{s - 1}$$and $$ Q_{r, ps} = \prod _{j = 1} ^r \binom{jps - 1}{ps - 1}, $$it follows from $$ (1 + x)^{jps - 1} \equiv (1 + x^p )^{js - 1}(1 + x)^{p - 1}\,\, \mod \,\, p$$ that $$ \binom{jps - 1}{ps - 1} \equiv \binom{js - 1}{s - 1} \,\, \mod \ p,$$whence the result. \section*{Practice}\addcontentsline{toc}{section}{Practice}\markright{Practice}\begin{multicols}{2}\columnseprule 1pt \columnsep 25pt\multicoltolerance=900 \begin{pro} Find a four-digit number which is a perfect square such that its first two digits are equal to each other and its last two digits are equal to each other.\end{pro} \begin{pro} Find all integral solutions of the equation $$ \sum _{k = 1} ^x k! = y^2 .$$ \end{pro} \begin{pro} Find all integral solutions of the equation $$\sum _{k = 1} ^x k! = y^z .$$ \end{pro} \begin{pro}[USAMO 1985] Determine whether there are any positive integral solutions to the simultaneous equations $$ x_1 ^2 + x_2 ^2 + \cdots + x_{1985} ^2 = y^3 ,$$ $$ x_1 ^3 + x_2 ^3 + \cdots + x_{1985} ^3 = z^2 $$with distinct integers $x_1 , x_2 , \ldots , x_{1985}$.\end{pro} \begin{pro} Show that the Diophantine equation $$ \dfrac{1}{a_1} + \dfrac{1}{a_2} + \ldots + \dfrac{1}{a_{n - 1}} + \dfrac{1}{a_n} + \dfrac{1}{a_1 a_2 \cdots a_n}$$ has at least one solution for every $n \in \BBN$. \end{pro} \begin{pro}[AIME 1987] Find the largest possible value of $k$ for which $3^{11}$ is expressible as the sum of $k$ consecutive positive integers. \end{pro} \begin{pro}[AIME 1987] Let ${\mathscr M}$ be the smallest positive integer whose cube is of the form $n + r$, where $n \in \BBN , 0 < r < 1/1000$. Find $n$. \end{pro} \begin{pro} Determine two-parameter solutions for the ``almost'' Fermat Diophantine equations $$ x^{n - 1} + y^{n - 1} = z^{n},$$$$ x^{n + 1} + y^{n + 1} = z^{n},$$$$ x^{n + 1} + y^{n - 1} = z^n .$$\end{pro} \begin{pro}[AIME 1984] What is the largest even integer which cannot be written as the sum of two odd composite numbers? \end{pro} \begin{pro} Prove that are infinitely many nonnegative integers $n$ which cannot be written as $n = x^2 + y^3 + z^6$ for nonnegative integers $x, y, z$. \end{pro} \begin{pro} Find the integral solutions of $$ x^2 + x = y^4 + y^3 + y^2 + y.$$\end{pro} \begin{pro} Show that there are infinitely many integers $x, y$ such that $$ 3x^2 - 7y^2 = -1.$$ \end{pro} \begin{pro}Prove that \begin{enumerate}\item $$ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 -ab - bc - ca).$$ \item Find integers $a, b, c$ such that $1987 = a^3 + b^3 + c^3 - 3abc.$ \item Find polynomials $P, Q, R$ in $x, y, z$ such that $$ P^3 + Q^3 + R^3 - 3PQR = (x^3 + y^3 + z^3 - 3xyz)^2$$ \item Can you find integers $a, b, c$ with $1987^2 = a^3 + b^3 + c^3 - 3abc$? \end{enumerate}\end{pro} \begin{pro} Find all integers $n$ such that $n^4 + n + 7$ is a perfect square. \end{pro} \begin{pro} Prove that $1991^{1991}$ is not the sum of two perfect squares.\end{pro} \begin{pro} Find infinitely many integers $x > 1, y > 1, z > 1$ such that $$ x!y! = z!.$$ \end{pro} \begin{pro} Find all positive integers with $$ m^n - n^m = 1.$$\end{pro} \begin{pro} Find all integers with $$ x^4 - 2y^2 = 1.$$\end{pro} \begin{pro}Prove that for every positive integer $k$ there exists a sequence of $k$ consecutive positive integers none of which can be represented as the sum of two squares.\end{pro} \begin{pro}[IMO 1977] In a finite sequence of real numbers, the sum of any seven successive terms is negative, and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.\end{pro} \begin{pro} Determine an infinite series of terms such that each term of the series is a perfect square and the sum of the series at any point is also a perfect square.\end{pro} \begin{pro} Prove that any positive rational integer can be expressed as a finite sum of distinct terms of the harmonic series, $1, 1/2, 1/3, \ldots$.\end{pro} \begin{pro}[Wostenholme's Theorem] Let $p > 3$ be a prime. If $$ \dfrac{a}{b} = 1 + \dfrac{1}{2} + \dfrac{1}{3} + \cdots + \dfrac{1}{p - 1},$$then $p^2|a.$\end{pro} \begin{pro} Prove that the number of odd binomial coefficients in any row of Pascal's Triangle is a power of $2$. \end{pro} \begin{pro} Prove that the coefficients of a binomial expansion are odd if and only if $n$ is of the form $2^k - 1.$ \end{pro} \begin{pro} Let the numbers $c_i$ be defined by the power series identity $$ (1 + x + x^2 + \cdots + x^{p - 1})/(1 - x)^{p - 1} := 1 + c_1 x + c_2 x^2 + \cdots .$$ Show that $c_i \equiv 0 \ \mod \ p$ for all $i \geq 1.$\end{pro} \begin{pro} Let $p$ be a prime. Show that $$ \binom{p - 1}{k} \equiv (-1)^k \ \mod \ p$$ for all $0 \leq k \leq p - 1.$\end{pro} \begin{pro}[Putnam 1977] Let $p$ be a prime and let $a \geq b > 0$ be integers. Prove that $$ \binom{pa}{pb} \equiv \binom{a}{b}\,\, {\rm\mod \,\, } p.$$ \end{pro} \begin{pro}Demonstrate that for a prime $p$ and $k \in \BBN$, $$ \binom{p^k}{a} \equiv 0 \ \mod \ p,$$ for $0 < a < p^k$. \end{pro} \begin{pro} Let $p$ be a prime and let $k, a \in \BBN , 0 \leq a \leq p^k - 1.$ Demonstrate that $$ \binom{p^k - 1}{a} \equiv (-1)^a \ \mod \ p.$$\end{pro} \end{multicols} \renewcommand{\chaptername}{Index} \printindex \end{document}