= Integer congruence
{title2=$a\equiv b\pmod m\iff m\mid(a-b)$}
For a positive modulus $m$, integers $a,b$ are congruent when their difference is divisible by $m$. This is an <equivalence relation>: zero is divisible by $m$, changing sign preserves <divisibility>, and adding two divisible differences proves transitivity. Addition and multiplication respect it, because $(a+c)-(b+d)=(a-b)+(c-d)$ and $ac-bd=c(a-b)+b(c-d)$. Its <equivalence classes> are the residue classes used in <modular arithmetic>.
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