06/19/2020 (2011: Countdown Round, Problem 20)

Q: Three teenagers have integer ages, x, y, and z, in years. If the product of their ages is 4590, and they each have a different age, what is the sum of the three ages, in years?

A: 50

Because the product ends in 0, that means that it is a multiple of 5. The only number that is a multiple of 5 that is also a teenager age is 15. 4590 ÷ 15 = 306.

The sum of the digits of 306 is 9, so that means it is a multiple of 9. The only number that is a multiple of 9 that is also a teenager age is 18. 306 ÷ 18 = 17.

That means the ages are 15, 17, and 18. 15 + 17 + 18 = 50.