---
tag: 09S5SN35DJ
kind: definition
name: Least common multiple
aliases: [lcm, lowest-common-multiple]
msc: [11A05]
defines: 007NPQX34K
uses: [096PW7Q014]
sources:
  - url: https://en.wikipedia.org/wiki/Least_common_multiple
    locator: lead section
---

## Definition

Let $a$ and $b$ be nonzero integers.
A common multiple of $a$ and $b$ is an integer $m$ such that $a \mid m$ and $b \mid m$.
The least common multiple $\operatorname{lcm}(a, b)$ is the smallest positive common multiple of $a$ and $b$.
By convention, $\operatorname{lcm}(a, 0) = \operatorname{lcm}(0, a) = 0$ for each integer $a$.

## Remarks

The positive integer $|a b|$ is a common multiple, so $\operatorname{lcm}(a, b)$ exists and $\operatorname{lcm}(a, b) \le |a b|$.
Each common multiple of $a$ and $b$ is a multiple of $\operatorname{lcm}(a, b)$.
For example, $\operatorname{lcm}(4, 6) = 12$.
The [[product of the gcd and the lcm|0TDY6JZHKS]] is the product of the two numbers.

## Explanations

### L1 human

List the positive multiples of each number, and take the smallest number that is in both lists.
The multiples of $4$ are $4, 8, 12, 16, \dots$ and the multiples of $6$ are $6, 12, 18, \dots$.
Thus $\operatorname{lcm}(4, 6) = 12$. You use this number when you add $\tfrac{1}{4} + \tfrac{1}{6}$ with a common denominator.