Problem Solving

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?

(A)5
(B)6
(C)15
(D)16
(E)18

Any shortcut to this? Thanks!

joyseychow wrote: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?

(A)5
(B)6
(C)15
(D)16
(E)18

Any shortcut to this? Thanks!

I think the answer is D

I approached this problem with the following things in mind:
1. The length of any number would be highest if the number is made up of the first couple of prime numbers.. i.e. 2, 3 or 5..
2. We need to mostly focus on 2 here..

With the above in mind.. 2^9 = 512.. so let x = 512..

x + 3y < 1000 => y < 163 (approx).. Focusing more on even numbers as we want to increase the count of 2, we know 2^7 = 128

So the maximum possible length can be 9 +7 = 16..

this is option D

Thanks!