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\newtheorem{lemma}{Lemma}
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\begin{quote}\color{red}Solution:\quad 
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\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\C}{\mathbb{C}}

\begin{document}
\title{Math 101 Homework}
\author{Mary Radcliffe}
\date{due 1 Feb 2015}
\maketitle

Complete the following problems. Fully justify each response.

\begin{enumerate}

\item Suppose $a, b, c, d$ are integers such that 
\begin{itemize}
\item $a$ and $b$ share no factors.
\item $c$ and $d$ share no factors.
\item $a^2d^2+b^2c^2=b^2d^2$
\end{itemize}

Prove that $b$ and $d$ must both be odd.

\item In class we showed that $\Q$ is countable. Prove that this implies $\Q^2$ is also countable. 

\item Let $X$ be a set, and $S\subset X$. Explain (in your own words) what it means for $S$ to be dense in $X$.
\end{enumerate}









\end{document}