I'd like to know why $m.(\bar{a}+\bar{b})=m.\bar{a}+m.\bar{b}$, $m \in \mathbb{Z}$, where
$\bar{a}=\{x \in \mathbb{Z}:x=a+nq\}$
. , + are the congruence operations of multiplication and sum.
and $\bar{a},\bar{b} \in \mathbb{Z}_n$ where:
$\mathbb{Z}_n=\{\bar{0}\,...,\bar{n-1}\}$
I was thinking about proving $m\in \mathbb{Z}_n$ since I know that:
