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\newtheorem{prop}{Proposition}
\newtheorem{thm}{Theorem}
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\lhead{MATH 311-01}
\chead{Problem of the Day}
\rhead{YOUR NAME HERE}
\lfoot{}
\cfoot{Page \thepage\ of \pageref{LastPage}}
\rfoot{\textbf{due November 6, 2013}}
\begin{document}
\emph{Let $[a]_m$ represent the congruence class of $a$ modulo
  $m$. Prove that, for $a,b\in\mathbb Z$ and $m,n\in\mathbb N$ the
  following are equivalent (i.e. that each is true if and only if the
  others are):}
\begin{itemize}
\item \emph{$[a]_m\subseteq [b]_n$.}
\item \emph{$a\equiv b\pmod n$ and $a+m\equiv b\pmod n$.}
\item \emph{$n\mid m$ and $a\equiv b\pmod n$.}
\end{itemize}

\begin{prop}
  The following are equivalent:
  \begin{enumerate}
  \item $[a]_m\subseteq [b]_n$.
  \item $a\equiv b\pmod n$ and $a+m\equiv b\pmod n$.
  \item $n\mid m$ and $a\equiv b\pmod n$.
  \end{enumerate}
\end{prop}
\begin{proof}
  We shall prove that the statements imply each other circularly;
  i.e. that (1)$\rightarrow$(2)$\rightarrow$(3)$\rightarrow$(1):

  {\bf (1)$\rightarrow$(2):}

  {\bf (2)$\rightarrow$(3):}

  {\bf (3)$\rightarrow$(1):}

\end{proof}
\end{document}
