# Definition:Lowest Common Multiple

From ProofWiki

## Definition

### Integral Domain

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

Then $l$ is the **lowest common multiple** of $a$ and $b$ 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 $a \mathrel \backslash m$ and $b \mathrel \backslash m$.

This $m$ is called the **lowest common multiple (LCM) of $a$ and $b$**, and denoted $\lcm \left\{{a, b}\right\}$.

## Also known as

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

The notation $\operatorname{lcm} \left\{{a, b}\right\}$ can be found written as $\left [{a, b} \right]$.

This usage is not recommended as it can cause confusion.

## Also see

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