# Category:Residue Classes

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This category contains results about Residue Classes.

Definitions specific to this category can be found in Definitions/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 the following 3 subcategories, out of 3 total.

### E

### R

## Pages in category "Residue Classes"

The following 8 pages are in this category, out of 8 total.