Definition:Greatest Common Divisor/Integers/General Definition

Definition
Let $S = \left\{{a_1, a_2, \ldots, a_n}\right\} \subseteq \Z$ such that $\exists x \in S: x \ne 0$ (that is, at least one element of $S$ is non-zero).

Then:
 * $\gcd \left({S}\right) = \gcd \left\{{a_1, a_2, \ldots, a_n}\right\}$

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

Also see

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