Least Common Multiple

number-theory

Definition

Least Common Multiple

The least common multiple of $a$ and $b$ is the smallest integer $d$ such that $a$ and $b$ both divide $d$.

$$\mathrm{lcm}(a,b)=\min \left\{d\in \mathbb{Z}\mid a|d\wedge b|d\right\}$$

Fundamental Theorem of Arithmetic

Using fundamental theorem of arithmetic, the least common multiple of $n,k\in \mathbb{Z}$ can be represented as:

$$\mathrm{lcm}(n,k)=\prod _i{p}_i^{\max (e_i(n),e_i(k))}$$

This can be useful when proving distribution laws for distributive lattices.