Hi Mathew. Let's say the numbers are A and B.

If the HCF is 3, then the only prime factor they share is 3 (because 3 itself is prime).

The LCM = HCF x (non-common-prime-factors from each of A and B).

By factoring 180, I find that 180 = 2 x 2 x 3 x 3 x 5.

Now I remove the HCF and have 2 x 2 x 3 x 5 left. I have to distribute these to A and B.

I know that A and B cannot both have factors of 2, because then the LCM would be 6 instead of 3 (3 x 2 = 6). So the 2 x 2 must be due to just one of the original numbers, so I'm left to distribute factors of 3, 4 and 5. I could give them all to one of the original numbers, or split them up. There are several ways of doing this:

A = 3 x 3, B = 3 x 4 x 5
or
A = 3 x 4, B = 3 x 3 x 5
or
A = 3, B = 3 x 3 x 4 x 5
or
A = 3 x 5, B = 3 x 3 x 4

Each of these will work. You could flip them around (switching B with A), but you'd still have the same two original numbers.

Cheers,
Stephen La Rocque and Penny Nom