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\begin{document}

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\noindent{\Large{\textsc{Number Theory\hfill Assignment 6.1  \\ Fall 2020 \hfill Due October 7}}}

\large

\vspace{1cm}


\begin{exercise}
Prove that 
\[
\begin{aligned}
n = &1678781422134741054423047947441055656115600301094736801 \backslash \backslash\\
      & 2425874667806297636227200527649707818860476378869522487
\end{aligned}
\]
is composite. [{\em Remark.}  This is a coding problem.  Please submit a well-documented copy of your code.  Feel free to use the programming language of your choice.] 
\end{exercise}

\begin{exercise}
Textbook exercise 5.2.16.
\end{exercise}

\begin{exercise}
Let $G$ be an abelian group.  Prove that for any $n \in \N$,
\[
G(n) = \{ a \in G \, | \, a^n = e \}
\]
is closed under multiplication and inversion in $G$.  Where do you need the fact that $G$ is abelian?
\end{exercise}

\begin{exercise}
Determine the 2-torsion subgroup of $\Z/n\Z$.
\end{exercise}

\begin{exercise}
Textbook exercise 5.3.10.
\end{exercise}

\end{document}
