# Lowest Common Multiple is Associative

## Theorem

Let $a, b, c \in \Z$.

Then:

$\lcm \left\{{a, \lcm \left\{{ b , c }\right\} }\right\} = \lcm \left\{{\lcm \left\{{a, b}\right\}, c}\right\}$

where $\lcm$ denotes the lowest common multiple.

## Proof

Follows directly from LCM from Prime Decomposition and Max is Associative