Definition:Congruence (Number Theory)/Integers/Modulo Operation

Definition
Let $m \in \Z_{>0}$. Let $\bmod$ be defined as the modulo operation:


 * $x \bmod m := \begin{cases}

x - m \left \lfloor {\dfrac x m}\right \rfloor & : m \ne 0 \\ x & : m = 0 \end{cases}$

Then congruence modulo $m$ is the relation on $\Z$ defined as:
 * $\forall x, y \in \Z: x \equiv y \pmod m \iff x \bmod m = y \bmod m$

The integer $m$ is called the modulus.

Also see

 * Congruence Modulo $z$ for real $z$


 * Equivalence of Definitions of Congruence


 * Congruence Modulo $m$ is Equivalence Relation