Definition:Greatest Common Divisor of Set of Integers/Definition 2

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).

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

is defined as the (strictly) positive integer $d \in \Z_{>0}$ such that:

where $\divides$ denotes divisibility.

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

Also see

 * Equivalence of Definitions of Greatest Common Divisor of Set of Integers