# LCM of Coprime Integers

## Theorem

Let $a, b \in \Z_{>0}$ be coprime integers.

Then:

$\lcm \set {a, b} = a b$

where $\lcm$ denotes the lowest common multiple.

## Proof

 $\displaystyle \lcm \set {a, b}$ $=$ $\displaystyle \frac {a b} {\gcd \set {a, b} }$ Product of GCD and LCM $\displaystyle$ $=$ $\displaystyle \frac {a b} 1$ Definition of Coprime Integers $\displaystyle$ $=$ $\displaystyle a b$

$\blacksquare$