The least common multiple (LCM) of two whole numbers $a$ and $b$ is the smallest whole number that is a multiple of both $a$ and $b.$ It is denoted as $\text{LCM}(a,b).$ The least common multiple of $a$ and $b$ is the smallest whole number that is divisible by both $a$ and $b.$ Some examples can be seen in the table below.

Numbers	Multiples of Numbers	Common Multiples	Least Common Multiple
$2$ and $3$	$$\begin{array}{rl}\text{Multiples of }2\text{:}& 2,4,6,8,10,12,\dots \\ \text{Multiples of }3\text{:}& 3,6,9,12,15,\dots \end{array}$$	$\underline{6},12,18,24,\dots$	$\text{LCM}(2,3)=6$
$8$ and $12$	$$\begin{array}{rl}\text{Multiples of }8\text{:}& 8,16,24,32,40,48,\dots \\ \text{Multiples of }12\text{:}& 12,24,36,48,\dots \end{array}$$	$\underline{24},48,72,96,\dots$	$\text{LCM}(8,12)=24$

A special procedure exists for finding the $\text{LCM}(a,b)$ of a pair of numeric expressions. The LCM of two relatively prime numbers is always equal to their product.

Coprimes	LCM
$3$ and $5$	$15$
$5$ and $4$	$20$
$4$ and $9$	$36$

Polynomials can also have a least common multiple. The LCM of two or more polynomials is the smallest multiple of both polynomials. In other words, the LCM is the smallest expression that can be evenly divided by each of the given polynomials.

Polynomials	Factor	LCM	Explanation
$$\begin{array}{c}4x^3\\ \text{and}\\ 6x{y}^2\end{array}$$	$$\begin{array}{c}{2}^2\cdot x^3\\ \text{and}\\ 2\cdot 3\cdot x\cdot y^2\end{array}$$	$12x^3{y}^2$	$12x^3{y}^2$ is the smallest expression that is divisible by both $4x^3$ and $6x{y}^2.$
$$\begin{array}{c}3x^3+3x^2\\ \text{and}\\ x^2+5x+4\end{array}$$	$$\begin{array}{c}3\cdot x^2\cdot (x+1)\\ \text{and}\\ (x+1)(x+4)\end{array}$$	$3x^2(x+1)(x+4)$	$3x^2(x+1)(x+4)$ is the smallest expression that is divisible by both $3x^3+3x^2$ and $x^2+5x+4.$

Finding the LCM of polynomials requires identifying the factors with the highest power that appear in each polynomial.