Bob and mark talk about their families. bob says he has 3 kids, the product of their ages is 72. he gives another clue: the sum of the ages of his children. mark points out that there is still not enough information to accurately guess. finally, bob says, "my youngest child called justice." mark can then correctly determine the ages of bob's children. what are the ages?

I have answered this question before. 

Given:
Bob has three kids whose ages has the product of 72. I will only be able to give you a set of 3 numbers to represent the ages. Had the sum of the ages been given, the specific ages would be provided.

I just did a prime factorization of 72.

72 ÷ 2 = 36
36 ÷ 2 = 18
18 ÷ 2 =   9
  9 ÷ 3 =   3
  3 ÷ 3 =   1

1 x 2 x 2 x 2 x 3 x 3 = 72   ⇒ 2³ x 3²

There are a lot of possible combinations, here are a few.

1 x 8 x   9 = 72      ⇒ 1 + 8 + 9 = 18
1 x 4 x 18 = 72      ⇒ 1 + 4 + 18 = 23
2 x 4 x   9 = 72      ⇒ 2 + 4 + 9 = 15