Logan has 4 teen age children of different ages. The product of the ages is 67184. Find the sum of the ages.

Hello!

I suppose that the ages are integer numbers, not fractional. Also teen age means integers from 13 to 19 (inclusive).

Let's decompose (factor) the given product, 67184. Try the factors at the given interval.

1) 67184 is divisible by 13, 67184 = 13*5168. Decompose the quotient.

2) 5168 isn't divisible by 14 and 15 but divisible by 16, 5168 = 16*323.

3) 323 is divisible by 17 and the quotient is 19. Hooray!

So we obtained that 67184 = 13*16*17*19. The only non-prime factor is 16, but we cannot divide it between other factors in a such way that they remain "teen age". Therefore this decomposition is the unique "teen" decomposition.

The answer for the question is 13 + 16 + 17 + 19 = 65.