# Definition:Lowest Common Multiple/Integers/Definition 1

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## Definition

For all $a, b \in \Z: a b \ne 0$, there exists a smallest $m \in \Z: m > 0$ such that $a \divides m$ and $b \divides m$.

This $m$ is called the **lowest common multiple of $a$ and $b$**, and denoted $\lcm \set {a, b}$.

## Warning

Note that unlike the GCD, where *either* of $a$ or $b$ must be non-zero, for the LCM *both* $a$ and $b$ must be non-zero.

Hence the stipulation:

- $a b \ne 0$

## 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

- Results about
**Lowest Common Multiple**can be found**here**.

## Sources

- 1971: George E. Andrews:
*Number Theory*... (previous) ... (next): $\text {2-2}$ Divisibility: Exercise $4$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $1$: Properties of the Natural Numbers: $\S 23 \gamma$ - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 7$: Example $7.8$