Let $M$ be a commutative monoid with cancellation, $a$,$b \in M$, and $d$ a $\text{gcd}(a,b)$. Then, $\frac{ab}{d}$ is a $\text{lcm}(a,b)$.
It is obvious that such a fraction is a common multiple of $a$ and $b$, but how to show that it divides all other common multiples?
