Definition:Greatest Common Divisor/Integers/Definition 2

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 (strictly) positive integer $d \in \Z_{>0}$ such that:
 * $(1): \quad d \divides a \land d \divides b$
 * $(2): \quad c \divides a \land c \divides b \implies c \divides d$

where $\divides$ denotes divisibility.

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

When $a = b = 0$, $\gcd \set {a, b}$ is undefined.

Also see

 * Equivalence of Definitions of Greatest Common Divisor