---
tag: 1TRB7CW647
kind: proof
name: Proof that the lcm of coprime integers is their product
msc: [11A05]
proves: 07CZYE06ZJ
uses: [0GFKP8F8AG, 09S5SN35DJ, 0TDY6JZHKS]
sources:
  - url: https://en.wikipedia.org/wiki/Least_common_multiple
---

## Proof

1. Let $a$ and $b$ be natural numbers that are [[coprime|0GFKP8F8AG]]. By definition, $\gcd(a, b) = 1$.
2. By the [[product of the gcd and the lcm|0TDY6JZHKS]], $\gcd(a, b) \cdot \operatorname{lcm}(a, b) = a b$.
3. Substituting $\gcd(a, b) = 1$ gives $\operatorname{lcm}(a, b) = a b$, where $\operatorname{lcm}$ is the [[least common multiple|09S5SN35DJ]].

## Idea

The gcd times the lcm is always the product. When the gcd is $1$, nothing is left to divide out, so the lcm is the product itself.