Remark 1

The relation $x \equiv y \pmod{a\mathbf{Z}}$ is usually written $x \equiv y \pmod{a}$, or simply $x \equiv y (a)$, and is read "x and y are congruent modulo a"; it means that $x - y$ is an integer multiple of $a$. When $a$ is an integer, the relation induced on $\mathbf{Z}$ by this relation is precisely congruence modulo $a$; this justifies the notation.