Definition:Greatest Common Divisor/Integers/General Definition

Definition
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 the greatest common divisor of $S$:
 * $\gcd \paren S = \gcd \set {a_1, a_2, \ldots, a_n}$

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

where $\divides$ denotes divisibility.

By convention:
 * $\map \gcd \O = 1$

Also see

 * Greatest Common Divisor is Associative for a justification of this construction.