\begin{definition}[Definition:Congruence (Number Theory)/Remainder after Division]
Let $z \in \R$.
We define a relation $\RR_z$ on the set of all $x, y \in \R$:
:$\RR_z := \set {\tuple {x, y} \in \R \times \R: \exists k \in \Z: x = y + k z}$
This relation is called '''congruence modulo $z$''', and the real number $z$ is called the '''modulus'''.
When $\tuple {x, y} \in \RR_z$, we write:
:$x \equiv y \pmod z$
and say:
:'''$x$ is congruent to $y$ modulo $z$'''.
Similarly, when $\tuple {x, y} \notin \RR_z$, we write:
:$x \not \equiv y \pmod z$
and say:
:'''$x$ is not congruent''' (or '''incongruent''') '''to $y$ modulo $z$'''.
\end{definition}