Definition:Lowest Common Multiple/Integers/General Definition

Definition
Let $S = \set {a_1, a_2, \ldots, a_n} \subseteq \Z$ such that $\ds \prod_{a \mathop \in S} a = 0$ (that is, all elements of $S$ are non-zero).

Then the lowest common multiple of $S$:
 * $\map \lcm S = \lcm \set {a_1, a_2, \ldots, a_n}$

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

where $\divides$ denotes divisibility.

Also see

 * Lowest Common Multiple is Associative for a justification of this construction.