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\begin{document}




{\bf Name:} \underline{\mbox{\hspace*{7cm}}}
\hfill{\bf Instructor:} Jeffrey Diller

\begin{center}
{\bf Math 223: Introduction to Mathematical Reasoning}
\\
{\bf Spring Semester 2004}
\\
{\bf Exam 2}
\\
{\bf Friday, April 3}
\end{center}
%\vspace*{.4cm}

\begin{tabular}{p{18cm}}\hline
This\rule[0cm]{0cm}{.8cm} examination contains 6 problems.  Counting
the front cover and blank pages, the exam consists of 7 sheets
of paper.  No calculators allowed.
\\[1.5ex]
\hline
\end{tabular}

\begin{center} {\bf Scores}

\begin{tabular}{|c|c|c|}\hline
Question & Possible\quad &\ Actual\quad\rule[-.3cm]{0cm}{0.9cm} \\ \hline\hline
  1      & 18   &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  2      & 32   &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  3      & 10   &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  4      & 15   &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  5      & 10   &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  6      & 15   &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
Total  & 100\mbox{\  \ } & \rule[-.3cm]{0cm}{0.9cm} \\ \hline
\end{tabular}
\end{center}

\vspace*{.5cm}
\begin{center}     {\bf GOOD LUCK}\end{center}
\newpage

\begin{problems}

\item Do each of the following (6 points each).  
\begin{subproblem}
\item What does the expression `$a \equiv b\mod n$' mean?  In
particular, what must $a$, $b$, and $n$ be for the expression to be meaningful?
\vspace*{1.5in}

\item Define \emph{relation}.
\vspace*{1.5in}

\item State Fermat's Little Theorem.
\vspace*{1.5in}

\end{subproblem}
\newpage

\item Four of the following seven assertions are false.  In the space
at the bottom of this page, identify three of them and give counterexamples.
Note that you do not have to justify your counterexamples---only present 
them. (8 points each)  
\begin{subproblem}
\item Given $a,b,c\in\N$ such that $a|bc$ it follows that $a|b$ or $a|c$. 
\item Given $a,b\in\Z$, there exists $m,n\in\Z$ such that 
      $am + bn = \gcd(a,b)$.
\item Every $\bar a\in \Z_5 - \{\bar 0\}$ has a multiplicative inverse.
\item If $a,b,k\in\Z$ and $k$ divides an integer combination of $a$ and
$b$, then $k|a$ and $k|b$.
\item If $2x\equiv 0\mod 4$, then $x \equiv 0\mod 4$.
\item If $a,b,k\in\Z$ are integers such that $k$ divides $a+b$ and $a$, then 
$k$ divides $b$.
\item Let us say that two integers are related if the difference between
the larger and the smaller of the two is prime.  Then this defines an 
equivalence relation on $\Z$.
\end{subproblem}
\newpage
\item 
Find the closest number $x\in\Z$ to 100 such that 
$$
x \equiv 1 \mod 2,\quad
x \equiv 1 \mod 3,\quad
x \equiv 5 \mod 7.
$$
Note that $x$ can be either greater or less than 100.
(10 points)
\newpage

\item 
Use the Euclidean algorithm to compute $k = \gcd(187,255)$ and to express
$k$ as an integer combination of $187$ and $255$.
(15 points)
\newpage

\ignore{\item 
Show that if $k|a$ and $k|b$, then $k$ divides any integer combination
of $a$ and $b$.}

\item 
Given $a,b,n\in\N$ such that $\gcd(a,b)=1$, show that $\gcd(na,nb) = n$.
(10 points)
\newpage

\item 
Prove that if $a,b\in\Z$ are integers such that $4 | (a^2+b^2)$, then
$a$ and $b$ are even.
(15 points)
\end{problems}

\end{document}

