---
tag: 0TPW25CBN6
kind: definition
name: Congruence modulo n
aliases: [congruence, modular-congruence]
msc: [11A07]
defines: 0WAY5WX8NY
uses: [096PW7Q014]
sources:
  - url: https://en.wikipedia.org/wiki/Modular_arithmetic
    locator: Congruence
---

## Definition

Let $n$ be a positive integer, and let $a$ and $b$ be integers.
The integer $a$ is congruent to $b$ modulo $n$ when $n$ [[divides|096PW7Q014]] $a - b$.
The notation is $a \equiv b \pmod{n}$. The integer $n$ is the modulus of the congruence.

## Remarks

Equivalently, $a \equiv b \pmod n$ if and only if $a$ and $b$ have the same remainder on division by $n$, that is, $a \bmod n = b \bmod n$.
Congruence modulo $n$ is an equivalence relation on the integers. Its classes are the [[residue classes|0QRV8FPK5M]] modulo $n$.
For example, $38 \equiv 14 \pmod{12}$, because $12 \mid 24$.
Some texts allow each integer $n$ as a modulus. Then congruence modulo $0$ is equality, and congruence modulo $-n$ is congruence modulo $n$.

## Explanations

### L1 human

Two integers are congruent modulo $n$ when they differ by a multiple of $n$.
On a $12$-hour clock, $14$ o'clock and $2$ o'clock show the same time, because $14 \equiv 2 \pmod{12}$.