Modular arithmetic

Suppose an integer $n \in {\bf Z} - \{ 0\} $ is fixed. A so-called congruence is defined on ${\bf Z}$ by $a \equiv b \operatorname{mod} n$ if $n$ divides $b-a$.

Theorem. Congruence is an equivalence relation.

The set of all equivalence classes of ${\bf Z}$ under this equivalence relation is denoted by ${\bf Z}_n$.

Theorem. ${\bf Z}_n$ is a group, as the quotient group: $${\bf Z}_n={\bf Z} / n{\bf Z.}$$