Congruence modulo n

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 $a - b$ . 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 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 $n$ .

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Divisibility definition
Congruence modulo n definition

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Tag	`0TPW25CBN6`
URI	`m2://0TPW25CBN6`
Aliases	`congruence`, `modular-congruence`
Kind	definition
Writing	W0 stub
MSC	11A07
Defines	Congruence modulo n
Uses	Divisibility

Congruence modulo n

Definition

Remarks

Reading path

Where this leads

Further reading

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