# 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*.

## Also see

- Greatest Common Measure of Commensurable Magnitudes where its existence is proven.