# Category:Definitions/Residue Classes

This category contains definitions related to Residue Classes.

Related results can be found in Category:Residue Classes.

Let $m \in \Z$.

Let $\mathcal R_m$ be the congruence relation modulo $m$ on the set of all $a, b \in \Z$:

- $\mathcal R_m = \set {\tuple {a, b} \in \Z \times \Z: \exists k \in \Z: a = b + k m}$

We have that congruence modulo $m$ is an equivalence relation.

So for any $m \in \Z$, we denote the equivalence class of any $a \in \Z$ by $\eqclass a m$, such that:

\(\displaystyle \eqclass a m\) | \(=\) | \(\displaystyle \set {x \in \Z: a \equiv x \pmod m}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \set {x \in \Z: \exists k \in \Z: x = a + k m}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \set {\ldots, a - 2 m, a - m, a, a + m, a + 2 m, \ldots}\) |

The equivalence class $\eqclass a m$ is called the **residue class of $a$ (modulo $m$)**.

## Subcategories

This category has only the following subcategory.

### R

## Pages in category "Definitions/Residue Classes"

The following 15 pages are in this category, out of 15 total.