Lcm of coprime integers

Statement

Let $a$ and $b$ be natural numbers that are coprime. Then $\operatorname{lcm}(a, b) = a b$.

Remarks

This follows from the product of the gcd and the lcm, because $\gcd(a, b) = 1$. The converse is also true when $a$ and $b$ are positive: if $\operatorname{lcm}(a, b) = a b$, then $a$ and $b$ are coprime. For example, $\operatorname{lcm}(8, 15) = 120$.

Explanations

L1 human

When $a$ and $b$ have no common factor, a multiple of both must contain all the factors of $a$ and all the factors of $b$. No factor can serve both numbers at the same time, so the smallest common multiple is $a b$.