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# Modular arithmetic

From Mathematics Is A Science

Jump to navigationJump to searchSuppose 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:
$${\bf Z}_n={\bf Z} / n{\bf Z.}$$