\documentclass[11pt]{amsart} \usepackage{graphicx} \usepackage{amssymb} \textwidth = 6.5 in \textheight = 10.3 in \oddsidemargin = 0.0 in \evensidemargin = 0.0 in \topmargin = -1.0 in \headheight = 0.0 in %\headsep = 0.1 in \parskip = 0.2in \parindent = 0.0in \newcommand{\pchoose}[2]{\begin{pmatrix}#1\\ #2\end{pmatrix}} \newcommand{\bchoose}[2]{\begin{bmatrix}#1\\ #2\end{bmatrix}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%% Young diagrams, grace a Francois Bergeron \newdimen\squaresize \squaresize=10pt \newdimen\thickness \thickness=0.4pt \def\square#1{\hbox{\vrule width \thickness \vbox to \squaresize{\hrule height \thickness\vss \hbox to \squaresize{\hss#1\hss} \vss\hrule height\thickness} \unskip\vrule width \thickness} \kern-\thickness} \def\vsquare#1{\vbox{\square{$#1$}}\kern-\thickness} \def\blk{\omit\hskip\squaresize} \def\noir{\vrule height\squaresize width\squaresize}% %\def\noir{\gray{\vrule height\squaresize width\squaresize}}% \def\blkk{\omit} \def\young#1{ \vbox{\smallskip\offinterlineskip \halign{&\vsquare{##}\cr #1}}} \def\thisbox#1{\kern-.09ex\fbox{#1}} \def\downbox#1{\lower1.200em\hbox{#1}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title{Homework \#5} \author{Date: March 4, 2020 : Due: March 18, 2020 } \begin{document} \maketitle In the following problems provide a complete explanation of why it is true \begin{enumerate} \item Let $n$ be an integer. Justify the following statements. \begin{enumerate} \item The last digit of $n$ is even if and only if $n$ is divisible by $2$. \item The last two digits of $n$ are divisible by $4$ if and only if $n$ is divisible by $4$. \item The last three digits of $n$ are divisible by $8$ if and only if $n$ is divisible by $8$. \item The last $k$ digits of $n$ are divisible by $2^k$ if and only if $n$ is divisible by $2^k$. \end{enumerate} \vskip .3in \item Let $n$ be an integer. Justify the following statements. \begin{enumerate} \item The integer $n$ is divisible by $3$ if and only if the sum of the digits is divisible by $3$. \item The integer $n$ is divisible by $9$ if and only if the sum of the digits is divisible by $9$. \end{enumerate} \vskip .3in \item What is the last nonzero digit at the end of $10!$~? What is the last nonzero digit at the end of $100!$ ? What is the last nonzero digit at the end of $1,\!000,\!000!$ ? Describe a procedure for finding the last non-zero digit at the end of $n!$ for any $n$. Use that procedure to find the last non-zero digit for the factorial of your student id number. \end{enumerate} \end{document}