# Definition:Lowest Common Multiple/Integers/Definition 2

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## Definition

Let $a, b \in \Z$ be integers such that $a b \ne 0$.

Then the **lowest common multiple** of $a$ and $b$ is the (strictly) positive integer $m$ which satisfies:

- $(1): \quad a \divides m$ and $b \divides m$
- $(2): \quad $If there exists $c \in \Z_{>0}$ such that $a \divides c$ and $b \divides c$, then $m \le c$

where $\divides$ denotes divisibility.

## 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

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts*... (previous) ... (next): Introduction $\S 6$: The division process in $I$ - 1980: David M. Burton:
*Elementary Number Theory*(revised ed.) ... (previous) ... (next): Chapter $2$: Divisibility Theory in the Integers: $2.3$ The Euclidean Algorithm: Definition $2 \text - 4$