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\prof{Chris Bendel and Peter Cholak} \class{Math 222}
\quizdate{Wednesday, February 17} \pagestyle{empty}

\begin{document}


\heading

Be sure to carefully write up your answers.  It is suggested that you
first write out a draft of your proposed questions and then carefully
rewrite that draft to get your final version.  You do \emph{not} have
to write the answers on this sheet of paper.\\

\begin{problem}
  Find the multiplicative inverse of 56 in $\mathbb{Z}_{61}$.
\end{problem}

\begin{problem}
  Prove that if $p$ is a prime and $\alpha, \beta \in \sqrt[p]{1}$ and
  $\alpha \neq 1$ then there exists an integer $m$ such that $\alpha^m
  = \beta$.  (Hints: First, write $\alpha = \zeta^k$ and $\beta = \zeta^r$
  where $\zeta$ is the first $p$th roof of unity.  Second, do $k$ and $r$
  have multiplicative inverses in $\mathbb{Z}_{p}$?)
\end{problem}

\begin{problem}
  Find the coeffieient of the term $x^{13}y^{4}$ in $(x + y)^{17}$.
  Is this number divisible by $17$?
\end{problem}

\end{document}
             