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\headers{Math303}{Homework 7}{Due: Wed. Feb 28, 2024}

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\noindent
\textbf{Directions:} You may work to solve these problems in groups, but all written work must be your own.  Unless the problem indicates otherwise, all problems require some justification; a correct answer without supporting reasoning is not sufficient.  See ``Guidelines and advice'' on the course webpage for more information.

\begin{enumerate}

\item Prove that if $x$ is an odd integer, then $x^3$ is odd.

\item Prove that if $x$ and $y$ are integers and $x$ is even, then $xy$ is even.

\item Prove that if $n\in\ZZ$, then $5n^2 + 3n + 7$ is odd.  Hint: try cases.

\item An integer $p$ is \emph{prime} if $p\ge 2$ and the only positive divisors of $p$ are $1$ and $p$.  Prove that if $n$ is a positive integer, $n\ge 2$, and $n$ is not prime, then $2^n - 1$ is not prime.



\end{enumerate}
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