Modulo operation

Definition

Let $a$ and $n$ be integers with $n \neq 0$ . The integer $a \bmod n$ is the remainder $r$ of the division of $a$ by $n$ . Thus $a \bmod n$ is the unique integer $r$ such that $0 \le r < |n|$ and $a - r$ is a multiple of $n$ .

Remarks

For example, $17 \bmod 5 = 2$ and $(-17) \bmod 5 = 3$ . Many programming languages use a different convention. In them, the remainder has the sign of $a$ , so the result for $-17$ and $5$ is $-2$ . The Lean library Mathlib sets $a \bmod 0 = a$ .

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Divisibility definition
Division algorithm statement
Modulo operation definition

Tag	`04FHQSJCQ5`
URI	`m2://04FHQSJCQ5`
Aliases	`mod`, `remainder`
Kind	definition
Writing	W1 stated
MSC	11A07
Defines	Modulo operation
Uses	Division algorithm

Modulo operation

Definition

Remarks

Reading path

Further reading

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