Def/Congruent

=Criteria= Suppose that $R$ is a Euclidean domain, such as $\ZZ$ or $\RR[X]$ for example. Suppose that $n$ is a nonzero element of $R$, and $x,y \in R$.

Let $r$ be the remainder obtained when $x$ is divided by $n$. Let $s$ be the remainder obtained when $y$ is divided by $n$. Then $x \equiv y$, mod $n$, if and only if $r \equiv s$, mod $n$.

As a special case, when $R = \ZZ$, we find that two integers are congruent, mod $n$, if they have the same remainder when divided by $n$ (using only positive remainders between $0$ and $n-1$, as is traditional).

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