# Definition:Coprime

## Definition

### GCD Domain

Let $\struct {D, +, \times}$ be a GCD domain.

Let $U \subseteq D$ be the group of units of $D$.

Let $a, b \in D$ such that $a \ne 0_D$ and $b \ne 0_D$

Let $d = \gcd \set {a, b}$ be the greatest common divisor of $a$ and $b$.

Then $a$ and $b$ are **coprime** if and only if $d \in U$.

That is, two elements of a GCD domain are **coprime** if and only if their greatest common divisor is a unit of $D$.

### Euclidean Domain

Let $\struct {D, +, \times}$ be a Euclidean domain.

Let $U \subseteq D$ be the group of units of $D$.

Let $a, b \in D$ such that $a \ne 0_D$ and $b \ne 0_D$

Let $d = \gcd \set {a, b}$ be the greatest common divisor of $a$ and $b$.

Then $a$ and $b$ are **coprime** if and only if $d \in U$.

That is, two elements of a Euclidean domain are **coprime** if and only if their greatest common divisor is a unit of $D$.

### Integers

Let $a$ and $b$ be integers such that $b \ne 0$ and $a \ne 0$ (that is, they are both non-zero).

Let $\gcd \set {a, b}$ denote the greatest common divisor of $a$ and $b$.

Then $a$ and $b$ are **coprime** if and only if $\gcd \set {a, b} = 1$.

## Also known as

The statement **$a$ and $b$ are coprime** can also be expressed as:

**$a$ and $b$ are relatively prime****$a$ and $b$ are mutually prime****$a$ is prime to $b$**, and at the same time that**$b$ is prime to $a$**.

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**prime to each other**