Product of the gcd and the lcm

Builds on Greatest common divisor Least common multiple

Statement

Let $a$ and $b$ be natural numbers. Then $\gcd(a, b) \cdot \operatorname{lcm}(a, b) = a b$ , where $\gcd$ is the greatest common divisor and $\operatorname{lcm}$ is the least common multiple.

Remarks

For integers $a$ and $b$ of any sign, the same argument gives $\gcd(a, b) \cdot \operatorname{lcm}(a, b) = |a b|$ . Thus $\operatorname{lcm}(a, b) = a b / \gcd(a, b)$ when $a$ and $b$ are not both $0$ . This is the usual way to compute the lcm, with the Euclidean algorithm. For example, $\gcd(4, 6) = 2$ , $\operatorname{lcm}(4, 6) = 12$ and $2 \cdot 12 = 24 = 4 \cdot 6$ .

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Divisibility definition
Greatest common divisor definition
Least common multiple definition
Product of the gcd and the lcm statement

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Lcm of coprime integers

Tag	`0TDY6JZHKS`
URI	`m2://0TDY6JZHKS`
Kind	statement, proposition
Writing	W0 stub
MSC	11A05
Status	This copy of the site has no build data. Run `m2 status 0TDY6JZHKS`.
Uses	Greatest common divisor, Least common multiple

Product of the gcd and the lcm

Statement

Remarks

Reading path

Where this leads

Further reading

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