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.

$\blacksquare$

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {2-2}$ Divisibility: Exercise $9$