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 $d \mathrel \backslash a$ and $d \mathrel \backslash b$.

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.

Also see

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


 * 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 $d = \pm d'$.