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 known as

The lowest common multiple is also known as the least common multiple.

It is usually abbreviated LCM, lcm or l.c.m.

The notation $\lcm \set {a, b}$ can be found written as $\sqbrk {a, b}$.

This usage is not recommended as it can cause confusion.

Also see

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

• Results about the lowest common multiple can be found here.

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {2-4}$ The Fundamental Theorem of Arithmetic: Exercise $10$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): common multiple
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): common multiple