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Modular arithmetic
From Mathematics Is A Science
Revision as of 05:01, 22 February 2011 by imported>WikiSysop
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: $${\bf Z}_n={\bf Z} / n{\bf Z.}$$