Q:

Proving distributive law in $\mathbb{Z}_{n}$ for equivalence classes

Equivalence classes defined as $\overline{a} = \{ b \in \mathbb{Z} : b \equiv a \mod n \}$ .
I need to prove the distributive law in $\mathbb{Z}_{n}$.  So, if $\overline{a}$, $\overline{b}$, and $\overline{c}$ are arbitrary elements in $\mathbb{Z}_{n}$, then $\overline{a}*(\overline{b}+\overline{c}) = \overline{a}*\overline{b}+\overline{a}*\overline{c}$.
I've seen examples proving the distributive law with real numbers and understood, but I'm not sure if the same proof works with equivalence classes.

A:

The definition of the operations on $\mathbb{Z}_n$ are
$$
\bar{a}+\bar{b}=\overline{a+b}
\qquad
\bar{a}\,\bar{b}=\overline{ab}
$$
Then
$$
\bar{a}(\bar{b}+\bar{c})=
\bar{a}\overline{(b+c)}=
\overline{a(b+c)}=\overline{ab+ac}
$$
Can you finish?

