Clue one gives all possible combinations of three ages that multiply to 36, which are:
2 2 9
2 6 3
4 3 3
1 4 9
1 6 6
1 2 18
1 3 12
Clue two still doesn't give the insurance person enough information, meaning that there must be (at least) two combinations adding to the same number. This leaves us with the two combinations:
2 2 9
1 6 6
which both add to 13.
Then last clue narrows it down to 2 2 9 as this is the only combination with an eldest.
An unusual sales pitch
Why is it that you have not listed 1,1,36 as a possibility? Is it because someone who is 36 is not a child? To a mother, her children are always children, regardless of their age.
Great point!
Figuring out the age combinations took a while. Even after realizing, from clue #2 that at least 2 age sums had to be the same, I had to sleep on it for it to hit me. Even then, I missed the 1,1,36 combo. Good catch!
I imagine that dropping the 1 is a common flaw when doing the prime factorization. That is what makes this difficult.