Chapter 2.6, Problem 15E

Prove Theorem $2.30$.

Theorem 2.30 Multiplication in ${\mathbb{Z}}_n$

Consider the rule for multiplication in ${\mathbb{Z}}_n$ given by

$\left[a\right]\left[b\right]=\left[ab\right]$.

Multiplication as defined by this rule is a binary operation on ${\mathbb{Z}}_n$.

Multiplication is associative in ${\mathbb{Z}}_n$:

$\left[a\right]\left(\left[b\right]\left[c\right]\right)=\left(\left[a\right]\left[b\right]\right)\left[c\right].$

Multiplication is commutative in ${\mathbb{Z}}_n$:

$\left[a\right]\left[b\right]=\left[b\right]\left[a\right]$.

${\mathbb{Z}}_n$ has the multiplicative identity $\left[1\right]$.