# Definition:Greatest Common Divisor/Integers

## Definition

Let $a, b \in \Z: a \ne 0 \lor b \ne 0$.

Then there exists a greatest element $d \in \Z_{>0}$ such that $d \divides a$ and $d \divides b$.

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

Its existence is proved in Existence of Greatest Common Divisor.

### General Definition

This definition can be extended to any (finite) number of integers.

Let $S = \set {a_1, a_2, \ldots, a_n} \subseteq \Z$ such that $\exists x \in S: x \ne 0$ (that is, at least one element of $S$ is non-zero).

Then:

$\gcd \paren S = \gcd \set {a_1, a_2, \ldots, a_n}$

is defined as the largest $d \in \Z_{>0}$ such that $\forall x \in S: d \divides x$.

## Also known as

The greatest common divisor is also known as the highest common factor (abbreviated HCF or hcf) and written $\operatorname {hcf} \set {a, b}$ or $\operatorname {hcf} \tuple {a, b}$.

Alternatively, $\gcd \set {a, b}$ is written in some texts as $\tuple {a, b}$, but this notation can cause confusion with ordered pairs.

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

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

## Also see

Thus it can be seen that for any two GCDs $d$ and $d'$ we have that $d = \pm d'$.

• Results about the greatest common divisor can be found here.