Least common multiple

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 is the product of the two numbers.

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Divisibility definition
Least common multiple definition

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Tag	`09S5SN35DJ`
URI	`m2://09S5SN35DJ`
Aliases	`lcm`, `lowest-common-multiple`
Kind	definition
Writing	W1 stated
MSC	11A05
Defines	Least common multiple
Uses	Divisibility

Least common multiple

Definition

Remarks

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Further reading

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