# What is the least common multiple of #{120, 130, 144}?

May 18, 2016

$9360$

#### Explanation:

Let us start by finding the prime factorisations of each of the numbers:

$120 = 2 \times 2 \times 2 \times 3 \times 5$

$130 = 2 \times 5 \times 13$

$144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3$

So the smallest number that contains all of these factors in these multiplicities is:

$2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 13 = 9360$

If you don't have a calculator to hand, an easier way to do that last multiplication might be:

$2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 13$

$= 13 \times \left(3 \times 3\right) \times 2 \times 2 \times 2 \times \left(2 \times 5\right)$

$= 13 \times 9 \times 2 \times 2 \times 2 \times 10$

$= 13 \times \left(10 - 1\right) \times 2 \times 2 \times 2 \times 10$

$= \left(130 - 13\right) \times 2 \times 2 \times 2 \times 10$

$= \left(117 \times 2\right) \times 2 \times 2 \times 10$

$= \left(234 \times 2\right) \times 2 \times 10$

$= \left(468 \times 2\right) \times 10$

$= 936 \times 10$

$= 9360$