Greatest Common Divisor divides Lowest Common Multiple
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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 $9$