Definition:Lowest Common Multiple/Integers/Definition 1

Definition

For all $a, b \in \Z: a b \ne 0$, there exists a smallest $m \in \Z: m > 0$ such that $a \divides m$ and $b \divides m$.

This $m$ is called the lowest common multiple of $a$ and $b$, and denoted $\lcm \set {a, b}$.

Warning

Note that unlike the GCD, where either of $a$ or $b$ must be non-zero, for the LCM both $a$ and $b$ must be non-zero.

Hence the stipulation:

$a b \ne 0$

Also known as

The lowest common multiple is also known as the least common multiple.

It is usually abbreviated LCM, lcm or l.c.m.

The notation $\lcm \set {a, b}$ can be found written as $\sqbrk {a, b}$.

This usage is not recommended as it can cause confusion.

Also see

• Equivalence of Definitions of Lowest Common Multiple

• Results about Lowest Common Multiple can be found here.

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {2-2}$ Divisibility: Exercise $4$
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: Properties of the Natural Numbers: $\S 23 \gamma$
• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 7$: Example $7.8$