# Definition:Lowest Common Multiple

Jump to navigation
Jump to search

## Definition

### Integral Domain

Let $D$ be an integral domain and let $a, b \in D$ be nonzero.

$l$ is the **lowest common multiple** of $a$ and $b$ if and only if:

- $(1): \quad$ both $a$ and $b$ divide $l$
- $(2): \quad$ if $m$ is another element such that $a$ and $b$ divide $m$, then $l$ divides $m$.

### Integers

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 $\lcm \set {a, b}$.

## 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
**the lowest common multiple**can be found**here**.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**least**(or**lowest**)**common multiple** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**least common multiple**