# The LCM of two different numbers is 18, and the GCF is 6. What are the numbers?

##### 2 Answers

They must have the prime factors

#### Explanation:

The numbers themselves are 6 and 18.

#### Explanation:

If the LCM of the two numbers is 18, that means both numbers have to be factors of 18. So each number could be 1, 2, 3, 6, 9, or 18.

If the GCF of the two numbers is 6, that means both numbers are divisible by 6. So each number could be 6, 12, 18, 24, ... etc.

When we overlap these two restrictions, we see that the only values common to both sets are 6 and 18. So our possible pairs are

But waitâ€”only one of these pairs has both an LCM of 18 *and* a GCF of 6. That pair is

## Bonus:

In general, you may also be able to use the fact that the product of any two numbers will equal the product of

#6=2*3#

#15=3 * 5#

GCF: They both have just a 3 in common, so their GCF is 3.

LCM: The first number that's a multiple of both will need to be a multiple of 2, 3, and 5. The *least* of such multiples is

So we get

#6 times 15#

#=(2 * 3) * (3 * 5)#

#=color(red)2 * color(blue)3 * color(red)3 * color(red)5#

#=color(red)lcm(6,15) times color(blue)("gcf"(6, 15))#

This will always work. For 6 and 18, however, it was not necessary (nor really very useful), since the numbers themselves were 6 and 18, which just so happened to be the GCF and LCM already.