---
tag: 0CHQKZFK5S
kind: statement
role: theorem
name: Division algorithm
aliases: [euclidean-division, division-with-remainder]
msc: [11A05]
uses: [096PW7Q014]
sources:
  - url: https://en.wikipedia.org/wiki/Euclidean_division
    locator: Division theorem
same_as:
  wikidata: Q181054
  wikipedia: Euclidean division
---

## Statement

Let $a$ and $b$ be integers with $b \neq 0$.
Then there is exactly one pair of integers $(q, r)$ such that $a = b q + r$ and $0 \le r < |b|$.

## Remarks

The integer $q$ is the quotient and the integer $r$ is the remainder of the division of $a$ by $b$.
The remainder is $0$ if and only if $b$ [[divides|096PW7Q014]] $a$.
For example, $-7 = 3 \cdot (-3) + 2$, so the quotient of $-7$ by $3$ is $-3$ and the remainder is $2$.

## Explanations

### L1 human

You can share $a$ objects into groups of $b$ objects, with some objects left over.
The theorem says that the number of full groups and the rest are unique, if the rest is smaller than $|b|$.
For example, $17 = 5 \cdot 3 + 2$: you get $3$ groups of $5$, and $2$ objects stay.