Integer congruence ($a\equiv b(\mathrm{mod}m)⟺m\mid (a-b)$)

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.

For every prime $p\ge 5$, $\left(\genfrac{}{}{0ex}{}{2p-1}{p-1}\right)\equiv 1(\mathrm{mod}{p}^3)$. Equivalently, ${\sum }_{k=1}^{p-1}1/k\equiv 0(\mathrm{mod}{p}^2)$, with denominators interpreted through modular inverses.