# Definition:Greatest Common Divisor/Real Numbers

## Definition

Let $a, b \in \R$ be commensurable.

Then there exists a greatest element $d \in \R_{>0}$ such that:

$d \divides a$
$d \divides b$

where $d \divides a$ denotes that $d$ is a divisor of $a$.

This is called the greatest common divisor of $a$ and $b$ and denoted $\gcd \set {a, b}$.

## Also known as

The greatest common divisor is often seen abbreviated as GCD, gcd or g.c.d.

Some sources write $\gcd \set {a, b}$ as $\tuple {a, b}$, but this notation can cause confusion with ordered pairs.

The notation $\map \gcd {a, b}$ is frequently seen, but the set notation, although a little more cumbersome, can be argued to be preferable.

The greatest common divisor is also known as the highest common factor, or greatest common factor.

Highest common factor when it occurs, is usually abbreviated as HCF, hcf or h.c.f.

It is written $\hcf \set {a, b}$ or $\map \hcf {a, b}$.

The archaic term greatest common measure can also be found, mainly in such as Euclid's The Elements.