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\begin{document}
 
\baselineskip.525cm
 
\title[Abstract Algebra]
{Exam 1\\Math 3613 Section 001, Spring 2009}
 
\author[Weiping Li]{Instructor: Weiping Li}
 
\maketitle

{\bf Print Name and Student $\#$}
\vspace{4ex}
\medskip

{\bf SHOW WORK FOR CREDIT !!! MUST PROVIDE LOGICAL REASONS!!!}
\medskip

\begin{enumerate}
\item (10pts) State the well-ordering axiom.
\vspace{2in}

\item (15pts) Prove that a and c leave the same remainder when divided by a positive integer n if and only if $a-c = nk$ for some integer k.
\newpage

\item (10pts) If $a|bc$ and $(a, b) =1$, then prove that $a|c$.
\vspace{3in}

\item (20pts) Prove that $\sqrt{a}$ is rational if and only if $a$ is a perfect square ($a=n^2$ for some integer n)
\newpage

\item (15pts) If $p\geq 5$ is prime, prove that $p^2 +2$ is composite.
\vspace{3.5in}

\item (15pts) If $a, b, c, d\in Z$ are integers and $a = bc -d$, prove that $(a, b) = (b, d)$.
\newpage

\item (15pts) (a) If $a, b, u, v$ are integers and $au+bv=1$, prove that $(a, b) =1$.
\vspace{3.5in}

(b) Show by example that if $au+bv = d > 1$, then $(a, b)$ may not be $d$.

\end{enumerate}
\end{document}