%Next time a problem using Eisiteen's crition should appear here and a
%harder problem about integer solutions of ax+by =g.
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\prof{Peter Cholak and Juan Migliore} \class{Math 222 - Sample Exam 1}
\quizdate{Friday, February 23} \pagestyle{empty}

\begin{document}


\heading

The exam will cover everything we have covered in class except the
general solutions of a cubic, and will have format as
outlined below.\\

\begin{problem}(5 parts at 4 points each) {\bf Define} the following terms:\\
  a) $\sqrt[6]{4}$\quad (over the complex numbers).\\
  b) The {\em argument} of a complex number $z$.\\
  c) The {\em order} of an $n$th root of unity for a positive integer $n$.\\
  d) The {\em additive inverse} of a number $a$ in $\mathbb{Z}_n$ ($n$
  a positive integer).\\
  e) {\em Constructible}.
\end{problem}

\begin{problem}(5 parts at 2 points each) 
  Answer {\bf True} or {\bf False} - no work
  required:\\
  a) $\operatorname{arg}(1 + i) = \pi/4 + 2{\pi}n$ for any $
  n \in \mathbb{Z}$.\\
  b) For any $z \in \mathbb{C}$, $|z|^3 = 3|z|$.\\
  c) $3\cdot17\equiv 3 \quad (\operatorname{mod} 6)$.\\
  d) The coefficient of $a^7b^4$ in $(a + b)^{11}$ is 330.\\
  e) the possible orders of elements in $\mathbb{Z}_7$ are $1,2,3,4,6$.
\end{problem}

\noindent {\bf Part III:} Computational Problems - there will 
be four problems something {\em like} the following for a total of
40 points.

\begin{problem} Find all solutions to $x^5-32=0$. Are these
  solutions constructible? Why or why not? (You do not need to
  simplify your answer to the form $a + bi$.)
\end{problem}

\begin{problem} Find the multiplicative inverse of 14 in
$\mathbb{Z}_{59}$.
\end{problem}

\begin{problem} Prove that $n^{13} - n$ is divisible by $78$ for
  any positive integer $n$.
\end{problem}

\begin{problem}
  Does $266x+361y=57$ have integer solutions?  If so find them.  It
  not, why not?  (You must use the Euclidean algorithm as discussed in
class.)
\end{problem}

\begin{problem}
  Find all roots of $x^{4^{666666}} + x^2 +3$ in $\mathbb{Z}_5$.
\end{problem}
\begin{problem}
  Find all the primitive elements in $\mathbb{Z}_{19}$
\end{problem}


\noindent {\bf Part IV:} Proofs - {\em exactly} two of the following 
problems will appear on the exam. They'll be worth 15 points each. Remember that ``$a \equiv_n b$'' is 
another way of writing ``$a \equiv b \  (\hbox{mod } n)$''.
\\

\begin{problem}
  Show $1\cdot 2 + 2\cdot 3 + 3\cdot4 + \ldots + n(n+1) =
  \frac{n(n+1)(n+2)}3$.
\end{problem}

\begin{problem}
  The \emph{least common multiple} of $x$ and $y$ is the least
  positive number which is a multiple of both $x$ and $y$.  Prove that
  the least common multiple of any two positive integers $k$ and $m$
  is $\displaystyle\frac{km}{(k,m)}$.
\end{problem}


\begin{problem}
  Let $a$ and $b$ be nonzero integers such that $g = (a,b)$.  Prove
  that $\displaystyle\left(\frac{a}{g},\frac{b}{g}\right)=1$.
\end{problem}

\begin{problem}
  For any prime $p$, if $a^p \equiv_p b^p$ then $a^p \equiv_{p^2}
  b^p$. (Hint: use Proposition 5.3.  )
\end{problem}

%\begin{problem}
%  Let $p, p_1, p_2, p_3 \in \mathbb{Z}$ be prime numbers.  Use
%  Euclid's Lemma (Lemma 4.4) to prove that if $p$ divides $n =
%  p_1\cdot p_2\cdot p_3$, then $p$ equals one of $p_1$, $p_2$, or
%  $p_3$.
%\end{problem}

\begin{problem}
  Let $m$, $n$ and $k$ be integers such that $(m,n)=1$ and both $m$
  and $n$ divide $k$.  Then prove that the product $mn$ divides $k$.
\end{problem}

\begin{problem}
  Let $p$ be prime.  Show every imaginary (complex and not real) 
  $p$th root of unity is a primitive $p$th root of unity.
\end{problem}

\end{document}


