Hi Henry.
When solving these problems, I like to use the prime factorizations whenever that's reasonable.
The prime factorization of 45 is 5 x 3 x 3.
The prime factorization of 60 is 2 x 2 x 3 x 5.
Therefore the HCF (highest common factor) of these two numbers is the set of factors that appear in both: that's one 3 and one 5. So the HCF is indeed 15.
The LCM (lowest common multiple) is the HCF times the factors left over from the first operation. Since there is a 3 and 2 factors of 2 left, the LCM = 15 x 3 x 2 x 2 = 180.
So we agree that 45 and 60 are solutions. Now your question is: Is this a unique solution?
Let's try working backwards.
The LCM is 180. The prime factorization of 180 is 5 x 3 x 3 x 2 x 2. The HCF is 15. The prime factorization of 15 is 5 x 3. If we remove the HCF factors from the LCM factors (in effect, we are dividing LCM/HCF), we get 3 x 2 x 2. So the two numbers are 15 times (some subset of 3 x 2 x 2) and 15 times (the remaining factors from 3 x 2 x 2). For example, 15 x 3 and 15 x 2 x 2 = 45, 60. Another example would be to have all the factors in one of the numbers: 15 and 15 x 2 x 2 x 3 = 180. So the numbers 15 and 180 themselves are indeed a pair of numbers whose HCF is 15 and whose LCM is 180. You are right that this is not a unique solution.
Note however that 15 x 2 and 15 x 3 x 2 would be a division of the factors as well. But that makes 30 and 90  clearly the LCM is 90. Why?
The reason is that if one 2 went to the first number and the other 2 went to the second number, then they'd have that as a common factor, and so the HCF wouldn't be 15, it would be 30 (with remaining factor 3 only).
So you were smart to wonder about the uniqueness of your answer  it isn't always obvious without digging in and looking at the actual prime factors involved.
Hope this helps,
Stephen La Rocque.
