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\begin{document}

\begin{center}
\includegraphics[width=0.5in]{../ltr-logo.jpg}
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\noindent{\Large{\textsc{Number Theory\hfill Assignment 1.1  \\ Fall 2020 \hfill Due September 2}}}

\large

\vspace{1cm}


\begin{exercise}
Let $a,b,c \in \Z$.  Prove that if $a|b$ and $b|c$, then $a|c$.
\end{exercise}

\begin{exercise}
Let $a,b,c,d \in \Z$.  Prove that if $a|b$ and $c|d$, then $ac|bd$.
\end{exercise}

\begin{exercise}
Let $a, b \in \Z$ and suppose $a$ is  divisor of $b$.  Prove that if $a \ne 0$, then its complementary divisor is unique.
\end{exercise}

\begin{exercise}
Let $b \in \Z$, $b \ne 0$.  For $a \in \Z$ define $r(a)$ to be the remainder when $a$ is divided by $b$.  Prove that for all $a_1, a_2, a_3 \in \Z$ one has
\[
r(a_1\cdot r(a_2 a_3)) = r(r(a_1 a_2) \cdot a_3) = r(a_1 a_2 a_3).
\]
\end{exercise}

\end{document}
