\documentclass{article}

\usepackage{amsmath, amssymb, amsthm}
\usepackage[margin=0.5in]{geometry}

\newcommand{\scs}[1]{\begin{scshape}#1\end{scshape}}
%\newcommand{\inv}{^{-1}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\NN}{\mathbb{N}}

\theoremstyle{definition}
\newtheorem*{definition}{Definition}
\newtheorem{problem}{}

\newcommand{\prob}[1]{\begin{problem}#1\end{problem}}
\newcommand{\dfn}[1]{\begin{definition}#1\end{definition}}

\newcommand{\abs}[1]{\left| #1 \right|}

\begin{document} \pagestyle{empty} \begin{flushleft}

\scs{Name(s):} 

\scs{Math 301 \hfill First Proofs \hfill February 6, 2015}

\vspace{.1in}

\dfn{
Let $n, m \in \ZZ$.
\begin{enumerate}

\item $n$ is {\em even} if there is $a \in \ZZ$ such that $n=2a$.

\item $n$ is {\em odd} if there is $a \in \ZZ$ such that $n = 2a+1$.

\item {\em $m$ divides $n$} (written $m|n$) if there is $a\in \ZZ$ such that $ma = n$.

\item If $n \geq 2$ and the only positive divisors $n$ are $1$ and $n$, then $n$ is {\em prime}.

\item If $n \geq 2$ and $n$ is not prime, then $n$ is {\em composite}.

\end{enumerate}
}

\prob{
Let $a, b, c \in \ZZ$.
Prove that if $a|b$ and $a|c$, then $a|(b+c)$.
}

\vfill

\prob{
Let $a, b, c \in \ZZ$.
Prove that if $a|b$ and $a|(b+c)$, then $a|c$.
}

\vfill

\newpage

\prob{
Let $x,y \in \ZZ$.
\begin{itemize}

\item[a) ] Prove that if $x$ or $y$ is even, then $xy$ is even.

\vfill

\item[b) ] State the converse of ``if $x$ or $y$ is even, then $xy$ is even''.
Note that the converse has the same implicit quantifier(s) as the original statement.

\vspace{.5in}

\item[c) ] Is the converse true or false? (There is no need to prove it).

\vspace{.5in}

\item[d) ] It is also true that if $6|x$ or $6|y$, then $6|(xy)$.
Is the converse of this statement true?

\vspace{.5in}

\end{itemize}
}

\prob{
Prove that every odd integer is the difference of two squares (e.g.\@ $5=3^2-2^2$).
}

\vfill

\end{flushleft} 
\end{document}