Definition:Congruence (Number Theory)/Remainder after Division

Definition
Let $z \in \R$.

We define a relation $\mathcal R_z$ on the set of all $x, y \in \R$:
 * $\mathcal R_z := \left\{{\left({x, y}\right) \in \R \times \R: \exists k \in \Z: x = y + k z}\right\}$

This relation is called congruence modulo $z$, and the real number $z$ is called the modulus.

When $\left({x, y}\right) \in \mathcal R_z$, we write:
 * $x \equiv y \pmod z$

and say:
 * $x$ is congruent to $y$ modulo $z$.

Similarly, when $\left({x, y}\right) \notin \mathcal R_z$, we write:
 * $x \not \equiv y \pmod z$

and say:
 * $x$ is not congruent (or incongruent) to $y$ modulo $z$.

Also see

 * Equivalence of Definitions of Congruence