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\textbf{College of the Holy Cross, Fall Semester 2017} \\
\textbf{MATH 243 -- Mathematical Structures, section 2} \\
\textbf{Exam 2 -- November 2}

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\noindent\textbf{Your Name}: \underline{\phantom{Name Name Name Name Name}}

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\paragraph{Instructions} Please write your answers in the spaces
provided on the following pages, and show work on the test itself. 
{\it For possible partial credit, even if you cannot completely solve a problem, 
include definitions of terms involved, partial results you can do, etc.}  Use the 
back of the preceding page if you need more space for scratch work.



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\noindent\textbf{\noindent\textbf{Please do not write in the space below}}


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Problem & Points/Poss    \strut \\

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I  & \hfil/\  20\hfil \strut \\

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II  & \hfil/\  35\hfil \strut \\

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III  & \hfil/\  15\hfil \strut \\

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IV  & \hfil/\  15\hfil \strut \\
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V  & \hfil/\  15\hfil \strut \\

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Total   & \hfil/100\hfil \strut \\

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\noindent
I.  Let $f : \mathbb{Z}/29\mathbb{Z} \to \mathbb{Z}/29\mathbb{Z}$
be the mapping defined by $f([x]) = [x] + [12]$.  
\begin{enumerate}
\item[(A)] (10) Show that $f$ is injective.
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\item[(B)] (10) Is $f$ surjective?  Why or why not?
\end{enumerate} 
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\noindent
II.  
\begin{enumerate}
\item[(A)](20) Give a precise statement of the Division Algorithm
in $\mathbb{Z}$, and prove {\it both} the Existence and Uniqueness parts. 
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\item[(B)] (15) Use the Euclidean algorithm to find the integer
$d = \gcd(585,108)$ and express $d$ in the form 
$d = m\cdot 585 + n\cdot 108$
for some integers $m,n$.
\end{enumerate}
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III.  (10)  Let $a,b,c$ be integers. 
Show that if $\gcd(a,b) = 1$ and $a|(bc)$, then $a|c$.  
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IV. (15) Find a solution $x$ of the congruence 
$31 x \equiv 6 \bmod 64$ with $0 \le x < 64$.
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V.  (15) Construct the \emph{multiplication} table for $(\mathbb{Z}/12\mathbb{Z})^\times$.
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