Definition:Greatest Common Divisor/Integers/Definition 1

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

The greatest common divisor of $a$ and $b$ is defined as:


 * the greatest element $d \in \Z_{>0}$ such that $d \divides a$ and $d \divides b$

where $\divides$ denotes divisibility.

This is denoted $\gcd \set {a, b}$.

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

Also see

 * Equivalence of Definitions of Greatest Common Divisor