Definition:Congruence (Number Theory)

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 $z$.

Definition for Integers
The concept of congruence is usually considered in the integer domain.

Also see

 * Equivalence of Congruence Definitions


 * Congruence Modulo $m$ is Equivalence Relation

Historical Note
The concept of congruence modulo an integer was first explored by Carl Friedrich Gauss.

Linguistic Note
The word modulo comes from the Latin for with modulus, that is, with measure.