 Integer congruence

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.