# What is the least common multiple of the set of numbers 8, 12, 16, 36?

Sep 4, 2016

144

#### Explanation:

$8 = {2}^{3}$
$12 = {2}^{2} \cdot 3$
$16 = {2}^{4}$
$36 = {2}^{2} \cdot {3}^{2}$
So the lowest common multiple must be ${2}^{4} \cdot {3}^{2}$
Or 144

Sep 5, 2016

$144$

#### Explanation:

The first thing to notice is that:
8 is a factor of 16 and
12 is a factor of 36.

Therefore we do not need to consider 8 and 12 at all,

Find the LCM of 16 and 36.

Find the product of their prime factors.

$\textcolor{w h i t e}{\times \times \times} 16 = 2 \times 2 \times 2 \times 2 \textcolor{w h i t e}{\times \times \times} = {2}^{4}$
$\textcolor{w h i t e}{\times \times \times} 36 = 2 \times 2 \textcolor{w h i t e}{\times \times \times} \times 3 \times 3 = {2}^{2} \times {3}^{2}$

LCM = $\textcolor{w h i t e}{. . \times} = 2 \times 2 \times 2 \times 2 \times 3 \times 3 = {2}^{4} \times {3}^{2} = 144$

A multiple of 16 must include ${2}^{4}$
A multiple of 36 must include ${2}^{2} \mathmr{and} {3}^{2}$

The LCM must have the highest of each, hence ${2}^{4} \times {3}^{2}$