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\newtheorem{prop}{Proposition}
\newtheorem{lem}{Lemma}
\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 September 18, 2013}}
\begin{document}
\begin{enumerate}
\item \emph{We will use the following steps to prove that, for odd
    integers $a$ and $b$ and integer $n$, if $an$ is even, then $bn$
    is even.}
  \begin{enumerate}
  \item \emph{First, prove the lemma that for an odd integer $b$ and
      an integer $n$, if $n$ is even, then $bn$ is even.}

    \begin{lem}
      For an odd integer $b$ and an integer $n$, if $n$ is even, then
      $bn$ is even.
    \end{lem}
    \begin{proof}
      % Your justification here!
    \end{proof}
  \item \emph{Then, prove the lemma that for an odd integer $a$ and an
      integer $n$, if $an$ is even, then $n$ is even.}

    \begin{lem}
      For an odd integer $a$ and an integer $n$, if $an$ is even, then
      $n$ is even.
    \end{lem}
    \begin{proof}
      % Your justification here!
    \end{proof}
    
  \item \emph{Using the two previous lemmas, prove that for odd
      integers $a$ and $b$ and integer $n$, if $an$ is even, then $bn$
      is even.}

    \begin{thm}
      For odd integers $a$ and $b$ and integer $n$, if $an$ is even,
      then $bn$ is even.
    \end{thm}
    \begin{proof}
      % Your justification here!
    \end{proof}
  \end{enumerate}
  
\item \emph{Using those same two lemmas, prove that for an odd integer
    $k$ and an integer $n$, the quantity $kn$ is even if and only if
    $n$ is even.}

  \begin{prop}
    For an odd integer $k$ and an integer $n$, the quantity $kn$ is
    even if and only if $n$ is even.
  \end{prop}
  \begin{proof}
      % Your justification here!
  \end{proof}
\end{enumerate}
\end{document}
