Residue Classes form Partition of Integers

Theorem
Let $m \in \Z_{>0}$ be a (strictly) positive integer.

Let $\Z_m$ be the set of all residue classes modulo $m$:


 * $Z_m = \set {\eqclass 0 m, \eqclass 1 m, \dotsc, \eqclass {m - 1} m}$

Then $Z_m$ forms a partition of $\Z$.

Proof
By definition of the set of all residue classes modulo $m$, $Z_m$ is the quotient set of congruence modulo $m$:
 * $\Z_m = \dfrac \Z {\mathcal R_m}$

where $\mathcal R_m$ is the congruence relation modulo $m$ on the set of all $a, b \in \Z$:
 * $\mathcal R_m = \set {\tuple {a, b} \in \Z \times \Z: \exists k \in \Z: a = b + k m}$

By the Fundamental Theorem on Equivalence Relations, $Z_m$ is a partition of $\Z$.