# Definition:Lowest Common Multiple/Integers/General Definition

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## Definition

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

Then the **lowest common multiple** of $S$:

- $\lcm \paren S = \lcm \set {a_1, a_2, \ldots, a_n}$

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.

## Sources

- 1971: George E. Andrews:
*Number Theory*... (previous) ... (next): $\text {2-4}$ The Fundamental Theorem of Arithmetic: Exercise $10$