Definition:Integers Modulo m

The quotient set of congruence modulo $m$ is:

$$\mathbb{Z}_m = \frac {\mathbb{Z}} {\mathcal{R}_m} = \left\{{\left[\!\left[{0}\right]\!\right]_m, \left[\!\left[{1}\right]\!\right]_m, \ldots, \left[\!\left[{m-1}\right]\!\right]_m}\right\}$$

where:
 * $$\mathcal{R}_m$$ is the equivalence relation congruence modulo $m$;
 * $$\left[\!\left[{x}\right]\!\right]_m$$ is the congruence class of $x$ modulo $m$.

Thus there are $$m$$ different congruence classes modulo $m$.

From Congruence to an Integer less than Modulus, it follows that the set defined here is a complete repetition-free list of them.

It also follows from Quotient Set forms a Partition that the quotient set $$\mathbb{Z}_m$$ of congruence modulo $m$ forms a partition of $$\mathbb{Z}$$.