Question

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

Answer 1

They must have the prime factors `2xx3=6` in common.

Explanation:

`18=2xx3xx3` and `6=2xx3` would do.

Answer 2

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 `(6, 6)`, `(6, 18)`, or `(18, 18)`.

But wait—only one of these pairs has both an LCM of 18 and a GCF of 6. That pair is `(6, 18)`. So, this is our answer.

Bonus:

In general, you may also be able to use the fact that the product of any two numbers will equal the product of `("their lcm") times ("their gcf")`. Let's show why with an example pair: 6 and 15.

`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 `2*3*5=30`.

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.