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\noindent{\Large{\textsc{Number Theory\hfill Assignment 3  \\ Spring 2022 \hfill Due March 2}}}

\large

\vspace{1cm}


\begin{exercise}
Given $a \in \Z$ and nonempty $S, T \subseteq \Z$, define
\[
\begin{aligned}
S+T &= \{ s+t \, | s \in S,t \in T \},\\[5pt]
aS &= \{ as \, | \, s \in S \},
\end{aligned}
\]
which are both subsets of $\Z$ as well.  Throughout our work with subgroups of $\Z$, we made implicit use of the following identities regarding these operations:  for all $a,b \in \Z$ and all $S , T \subseteq \Z$ one has
\[
\begin{aligned}
a(S+T) &= aS + aT,\\[5pt]
a(bS) &= (ab)S.
\end{aligned}
\]
Prove these carefully using double-containment arguments.  Determine (with proof) whether or not $(a+b)S = aS + bS$ is also a valid identity.
\end{exercise}

\begin{exercise}
Let $S ,T ,U,V \subseteq \Z$.
\begin{itemize}

\item[\bf a.]  Prove that if $S \subseteq U$ and $a \in \Z$, then $aS \subseteq aU$.

\item[\bf b.] Prove that if $S \subseteq U$ and $T \subseteq V$, then $S+T \subseteq U + V$.

\end{itemize}

\end{exercise}

\begin{exercise}
Given nonzero $a_1, a_2, \ldots, a_r \in \Z$, define their {\em greatest common divisor} $(a_1, a_2, \ldots, a_r)$ to be the largest $a \in \N$ so that $a|a_i$ for all $i$.  Using the techniques of subgroups that we have introduced in class, without appealing to B\'ezout's lemma, prove that
\[
a_1 \Z + a_2 \Z + \cdots + a_r \Z = (a_1, a_2, \ldots , a_r) \Z.
\] 
Conclude that every common divisor of the $a_i$ must in fact divide $(a_1, a_2, \ldots, a_r)$.  
\end{exercise}

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