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\noindent{\Large{\textsc{Number Theory \hfill Assignment 5.1  \\ Fall 2023 \hfill Due September 27}}}

\large

\vspace{1cm}


Fix a modulus $n \in \N$.  Recall that for $a \in \Z$ the congruence class of $a \pmod{n}$ is
\[
a+n \Z = \{a + kn \, | \, k \in \Z \},
\]
which is simply the equivalence class of $a$ under congruence mod $n$.  By definition we have
\[
a \equiv b \!\!\!  \pmod{n} \,\,\, \Leftrightarrow \,\,\, a + n \Z = b + n \Z.
\]
Because every congruence class contains exactly one of the integers $0, 1, 2, \ldots, n-1$, the set of all equivalence classes (quotient space) is
\[
\Z/n\Z := \{ n \Z, 1 + n\Z, 2+n \Z, \ldots, (n-1) + a \Z \}.
\]
In the following exercises you will prove that $\Z/n \Z$ has the algebraic structure of a {\em ring}, so that modular arithmetic in $\Z$ is equivalent to ordinary arithmetic in $\Z/n\Z$.  We will take advantage of this ring structure to prove several important results in the theory of congruences.

\vspace{1cm}

\begin{exercise}
For $a+ n\Z, b + n\Z \in \Z/n\Z$, define their {\em sum} to be the congruence class of $a+b$, that is
\[
(a+ n\Z) + (b+ n \Z) := (a+b) + n\Z,
\]
and define their {\em product} to be the congruence class of $ab$, or
\[
(a+ n\Z) (b+ n \Z) := ab + n\Z.
\]
Use established properties of congruences to prove that these operations are {\em well-defined}.  In other words, prove that if $a+ n\Z = c+ n\Z$ and $b+ n\Z = d+ n\Z$, then
\[
(a+b) + n\Z = (c+d) + n\Z \,\,\, \text{ and } \,\,\, ab + n\Z = cd + n\Z.
\]
\end{exercise}

\vspace{-0.5cm}

\begin{exercise}
Show that $\Z/n\Z$ with the addition operation defined above is an abelian group.
\end{exercise}

\begin{exercise}
Show that $\Z/n\Z$ with the multiplication operation defined above is a commutative monoid.
\end{exercise}

\begin{exercise}
Show that multiplication of congruence classes in $\Z/n\Z$ distributes over addition of congruence classes.
\end{exercise}

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