\documentclass[12pt]{amsart}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage[all]{xy}
\voffset=-1in
\hoffset=-1in
\textheight=10in
\textwidth=6in
\begin{document}
\author{}
\title{Math 420-620\\Fall 2012\\Homework 1}
\date{}
\maketitle
\thispagestyle{empty}
\pagestyle{empty}



\centerline{\it Due Friday August 31, 2010.}

\centerline{}

\noindent 1. Let $k,m,n\in\mathbb{Z}$ be nonzero integers. 
\begin{itemize}
\item[a)] (5 pt) Show that $\text{gcd}(m,n)$ is a linear combination of $m$ and $n$ (that is, show that there are integers $a$ and $b$ such that $am+bn=\text{gcd}(m,n)$).
\item[b)] (5 pt) Show that if $\text{gcd}(k,m)=1$ and $\text{gcd}(k,n)=1$, then $\text{gcd}(k,mn)=1$.
\item[c)] (5 pt) Show that if $\text{gcd}(k,m)=1$ and $k$ divides $mn$, then $k$ divides $n$.
\end{itemize}


\centerline{}

\noindent 2. (5 pt) Let $m,n\in\mathbb{Z}$ be nonzero integers, $d=\text{gcd}(m,n)$ and $L=\text{lcm}(m,n)$. Show that $dL=mn$.


\centerline{}

\noindent 3. Let $A$ be a nonempty set and $\sim$ an equivalence relation on $A$.
\begin{itemize}
\item[a)] (5 pt) Show that the set of equivalence classes of $A$ under $\sim$ is a partition of $A$.
\item[b)] (5 pt) Show that if $\{A_i\}_{i\in \Lambda}$ is a partition of $A$, then there is an equivalence relation on $A$ such that the sets $\{A_i\}_{i\in \Lambda}$ are precisely the equivalence classes of $A$ under this relation.
\end{itemize}



\end{document}