Definition:Lowest Common Multiple

From ProofWiki
Jump to: navigation, search


Integral Domain

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


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 (LCM) 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.

The notation $\operatorname{lcm} \left\{{a, b}\right\}$ can be found written as $\left [{a, b} \right]$.

This usage is not recommended as it can cause confusion.

Also see

  • Results about Lowest Common Multiple can be found here.