Subject: HCF and LCM
Name: Henry
Who are you: Parent

Is there a unique solution to the question:
If the LCM and HCF of two numbers are 180 and 15 respectively, what are the two numbers?

I got 45 and 60. I got a feeling there are others.


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.