Definition:Congruence (Number Theory)/Integers

Definition

Let $m \in \Z_{> 0}$.

Then we define congruence modulo $m$ as the relation $\mathcal R_m$ on the set of all $a, b \in \Z$:

$\mathcal R_m = \left\{{\left({a, b}\right) \in \Z \times \Z: \exists k \in \Z: a = b + km}\right\}$

That is, such that $a$ and $b$ have the same remainder when divided by $m$.

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:

$\forall x, y \in \Z: x \equiv y \pmod m \iff x \bmod m = y \bmod m$

We also see that $a$ is congruent to $b$ modulo $m$ if their difference is a multiple of $m$:

Let $x, y \in \Z$.

Then $x$ is congruent to $y$ modulo $m$ iff their difference is an integral multiple of $m$:

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