Definition:Greatest Common Divisor/Integers

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

Then there exists a largest $d \in \Z_{>0}$ such that $d \mathop \backslash a$ and $d \mathop \backslash b$.

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

Its existence is proved in Existence of Greatest Common Divisor.

Largest here is, of course, formally defined as maximal with respect to the ordering $\le$.

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

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

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

The notation $\gcd \left({a, b}\right)$ is also seen, but the set notation, although arguably more cumbersome, can be argued to be preferable.


 * GCD iff Divisible by Common Divisor for an equivalent definition.

Also see

 * Elements of Euclidean Domain have Greatest Common Divisor where it is shown that any two GCDs of $a$ and $b$ are associates.

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