For any integer k > 1, the term “length of an integer” refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. For example, if k = 24, the length of k is equal to 4, since 24 = 2 × 2 × 2 × 3. If x and y are positive integers such that x > 1, y > 1, and x + 3y < 1000, what is the maximum possible sum of the length of x and the length of y?
Choose the correct choice from the following answers
A) 5 B) 6 C) 15 D) 16 E) 18

Rationale: basically the length of an integer is the sum of the powers of its prime factors . for example the length of 24 is 4 because 24 = 2 ^ 3 * 3 ^ 1 - - > 3 + 1 = 4 . given : x + 3 y < 1,000 . now , to maximize the length of x or y ( to maximize the sum of the powers of their primes ) we should minimize their prime bases . minimum prime base is 2 : so if x = 2 ^ 9 = 512 then its length is 9 - - > 512 + 3 y < 1,000 - - > y < 162.7 - - > maximum length of y can be 7 as 2 ^ 7 = 128 - - > 9 + 7 = 16 . answer : d .
Answer: D