---
tag: 0GFKP8F8AG
kind: definition
name: Coprime integers
aliases: [relatively-prime, mutually-prime]
msc: [11A05]
defines: 0VZ48N083F
uses: [02WF878BH1]
sources:
  - url: https://en.wikipedia.org/wiki/Coprime_integers
    locator: lead section
---

## Definition

Two integers $a$ and $b$ are coprime when their [[greatest common divisor|02WF878BH1]] is $1$, that is, when $\gcd(a, b) = 1$.

## Remarks

Equivalently, $a$ and $b$ are coprime when the only integers that divide both of them are $1$ and $-1$.
For example, $8$ and $15$ are coprime, but $8$ and $12$ are not, because $4$ divides both.
The integers $1$ and $-1$ are coprime to each integer. The integer $0$ is coprime only to $1$ and $-1$.
By [[Bézout's identity|0DQVHMYAGA]], $a$ and $b$ are coprime if and only if $a x + b y = 1$ for some integers $x$ and $y$.

## Explanations

### L1 human

Two numbers are coprime when they have no common factor except $1$.
The numbers do not need to be prime themselves: $9$ and $10$ are not prime, but they are coprime.
A fraction $a / b$ is in lowest terms exactly when $a$ and $b$ are coprime.