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\begin{document}

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\noindent{\Large{\textsc{Number Theory \hfill Assignment 1.1  \\ Fall 2023 \hfill Due August 30}}}

\large

\vspace{1cm}

\begin{exercise}
Let $a, b , c \in \Z$ with $a \ne 0$.  Use the fact that $\Z$ is a domain to prove that if $ab=ac$, then $b=c$.  This is called the {\em cancellation law} in $\Z$.  [{\em Suggestion.}  If $ab=ac$, then $ab-ac = 0$.  Now factor out $a$.]
\end{exercise}

\begin{exercise}
Let $a,b \in \Z$ with $b \ne 0$.  Prove that if $b = ac$ for some $c \in \Z$, then $c$ is unique.  [{\em Suggestion.}  Write $b = ac = ac'$ and use the preceding exercise.]
\end{exercise}

\begin{exercise}
Prove that divisibility in $\Z$ is transitive.  That is, show that if $a,b,c \in \Z$ with $a|b$ and $b|c$, then $a|c$.
\end{exercise}

\begin{exercise}
Let $a,b \in \Z$.  Show that if $ab=1$, then $a,b \in \{ \pm 1 \}$ and $a=b$.  [{\em Suggestion.}  Argue by contradiction.  Use the fact (proved in class) that if $a \ne 0$ and $|b|>1$, then $|ab|>|a|$.]
\end{exercise}

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