Definition:Lowest Common Multiple/Integers/Definition 2

From ProofWiki
Jump to navigation Jump to search


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.


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

  • Results about Lowest Common Multiple can be found here.