Upper Bound for Lowest Common Multiple

Theorem

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

Then:

$\lcm \set {a, b} \le \size {a b}$

where:

$\lcm \set {a, b}$ denotes the lowest common multiple of $a$ and $b$

Proof

By Product of GCD and LCM:

$\lcm \set {a, b} \times \gcd \set {a, b} = \size {a b}$

where:

$\gcd \set {a, b}$ denotes the greatest common divisor of $a$ and $b$.

By Existence of Greatest Common Divisor $\gcd \set {a, b}$ exists.

By definition of GCD, $\gcd \set {a, b} \in \Z_{>0}$.

Hence the result.

$\blacksquare$

Sources

• 1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $2$: Divisibility Theory in the Integers: $2.3$ The Euclidean Algorithm