Existence of Lowest Common Multiple/Proof 3

Theorem

Let $a, b \in \Z: a b \ne 0$.

The lowest common multiple of $a$ and $b$, denoted $\lcm \set {a, b}$, always exists.

Proof

Note that as Integer Divides Zero, both $a$ and $b$ are divisors of zero.

Thus by definition $0$ is a common multiple of $a$ and $b$.

Non-trivial common multiples of $a$ and $b$ exist.

Indeed, $a b$ and $-\paren {a b}$ are common multiples of $a$ and $b$.

Either $a b$ or $-\paren {a b}$ is strictly positive.

Let $S$ denote the set of strictly positive common multiples of $a$ and $b$.

By the Well-Ordering Principle, $S$ contains a smallest element.

This can then be referred to as the lowest common multiple of $a$ and $b$.

$\blacksquare$

Sources

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