FORTNIGHT'S PUZZLE 16

PUZZLE 16: 24th-30th May (to be e-mailed by the 31st May)
POSTED BY IES CAMPANAR

A NUMEROUS FAMILY
The product of my sons’ ages is 1664. The age of the eldest is the double of my youngest son’s age. I am 50 years old myself.How many sons have I got?How old are they?

SOLUTION
The factorial decomposition of 1664 is the product of 13 x 2 with exponent 7. Since 13 can be neither the youngest nor the eldest son, the solution is 2 with exponent 3 = 8, 13, and 2 with exponent 4 = 16. Therefore the three sons are: 8, 13 and 16 years old.