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\begin{document}

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\noindent{\Large{\textsc{Number Theory\hfill Assignment 1.2  \\ Fall 2020 \hfill Due September 2}}}

\large

\vspace{1cm}


\begin{exercise}
Prove that for all $n \in \N$, $15 | 2^{4n}-1$.  [{\em Suggestion:}  Factor the polynomial $X^m-1$.]

\end{exercise}

\begin{exercise}
Prove that the square of any integer has the form $3k$ or $3k+1$.  Use this to show that $3a^2 - 1$ is never a perfect square.
\end{exercise}

\begin{exercise}
If $n$ is odd, prove that $16 | n^4 + 4n^2 + 11$.  [{\em Suggestion:}  Use the fact that $11 = 16-5$.]
\end{exercise}

\begin{exercise}
Use the Euclidean algorithm to compute the following GCDs.
\begin{itemize}

\item[\bf a.]  $(143, 277)$

\item[\bf b.] $(306, 657)$

\item[\bf c.] $(272, 1479)$

\end{itemize}

\end{exercise}

\end{document}
