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\begin{document}

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\noindent{\Large{\textsc{Number Theory \hfill Assignment 2.1  \\ Fall 2023 \hfill Due September 6}}}

\large

\vspace{1cm}

\begin{exercise}
Let $a, b \in \Z$ with $a \ne 0$.  Use the Pigeonhole Principle to show that if we apply the Division Algorithm to divide the consecutive integers $b, b+1, b+2, \ldots, b+(|a|-1)$ by $a$, then every possible remainder occurs exactly once. 
\end{exercise}

\begin{exercise}
Textbook exercise 2.2.11. [{\em Suggestion.} Start by writing $n^4 + 4n^2 + 11 = (n^2+2)^2 + 7$.]
\end{exercise}

\begin{exercise}
Textbook exercise 2.3.12.
\end{exercise}

\begin{exercise}
Textbook exercise 2.3.14.
\end{exercise}

\end{document}
