Greatest Common Divisor divides Lowest Common Multiple

Theorem
Let $a, b \in \Z$ such that $a b \ne 0$.

Then:


 * $\gcd \set {a, b} \divides \lcm \set {a, b}$

where:
 * $\lcm$ denotes lowest common multiple


 * $\gcd$ denotes greatest common divisor.


 * $\divides$ denotes divisibility.

Proof
We have that:


 * $\gcd \set {a, b} \divides a$

and:
 * $a \divides \lcm \set {a, b}$

The result follows from Divisor Relation is Transitive.