# What is the LCM of 45, 28, and 150?

May 9, 2017

$L C M = 2 \times 2 \times 3 \times 3 \times 5 \times 5 \times 7 = 6300$

#### Explanation:

Write each number as the product of its prime factors. In this way you know exactly what you are working with, which factors are common and how many of each you need for the LCM.
Line up like factors under each other.

$\textcolor{w h i t e}{\ldots \ldots .} 45 = \textcolor{w h i t e}{\ldots \ldots \ldots . .} 3 \times 3 \times 5$
$\textcolor{w h i t e}{\ldots \ldots .} 28 = 2 \times 2 \textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots . .} \times 7$
$\textcolor{w h i t e}{\ldots \ldots} \underline{150 = 2 \textcolor{w h i t e}{\ldots \ldots \ldots .} 3 \textcolor{w h i t e}{\ldots \ldots} \times 5 \times 5 \text{ }}$
$\textcolor{w h i t e}{.} L C M = 2 \times 2 \times 3 \times 3 \times 5 \times 5 \times 7 = 6300$

Notice that in order for:

$28$ to be a factor, there must be two $2 s \mathmr{and} \text{one } 7$
$45$ to be a factor, there must be two $3 s$
$150$ to be a factor, there must be two $5 s$

The LCM is the product of all the factors, but without the duplicates.