Definition:Lowest Common Multiple/Integers/Definition 2

Definition

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

Then the lowest common multiple of $a$ and $b$ is the (strictly) positive integer $m$ which satisfies:

$(1): \quad a \divides m$ and $b \divides m$

$(2): \quad $If there exists $c \in \Z_{>0}$ such that $a \divides c$ and $b \divides c$, then $m \le c$

where $\divides$ denotes divisibility.

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

• 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 6$: The division process in $I$
• 1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $2$: Divisibility Theory in the Integers: $2.3$ The Euclidean Algorithm: Definition $2 \text - 4$