# Definition:Congruence (Number Theory)/Integer Multiple

## Definition

Let $z \in \R$.

Let $x, y \in \R$.

Then $x$ is congruent to $y$ modulo $z$ if and only if their difference is an integer multiple of $z$:

$x \equiv y \pmod z \iff \exists k \in \Z: x - y = k z$

## Notation

The relation $x$ is congruent to $y$ modulo $z$, usually denoted:

$x \equiv y \pmod z$

is also frequently seen denoted as:

$x \equiv y \ \left({\mathop {\operatorname{modulo} } z}\right)$

Some (usually older) sources render it as:

$x \equiv y \ \left({\mathop {\operatorname{mod.} } z}\right)$

## Linguistic Note

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