Definition:Lowest Common Multiple

Definition

Integral Domain

Let $D$ be an integral domain and let $a, b \in D$ be nonzero.

$l$ is the lowest common multiple of $a$ and $b$ if and only if:

$(1): \quad$ both $a$ and $b$ divide $l$

$(2): \quad$ if $m$ is another element such that $a$ and $b$ divide $m$, then $l$ divides $m$.

Integers

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 $\lcm \set {a, b}$.

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

• Definition:Greatest Common Divisor

• Results about the lowest common multiple can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): least (or lowest) common multiple
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): least common multiple
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): least common multiple
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): least common multiple