"Length of integer" (prime number properties)

[JohnRapp]

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

I'm hoping this post can provide alternative explanations to the OA explanation.

OA = [spoiler](D) 16[/spoiler]

[rijul007]

for a number to have maximum length.. it should have the highest no of prime factors not necessarily distinct

this number n should have max nuber of 2s as a prime factor

x>1
y>1

x = 2^a [a is the length of x]
y = 2^b

x+3y<1000
2^a + 3(2)^b <1000

highest value of 2^a that can fit into this is 512

2^a = 512
a = 9

3(2)^b <1000-512
2^b < 488/3
2^b < 162
highest value of 2^b can be 128
b = 7

Maximum sum of length of a and length of b = a + b =9+7 = 16


Option D

[rijul007]

for a number to have maximum length.. it should have the highest no of prime factors not necessarily distinct

a number should have max nuber of 2s as a prime factor for the max length..

x>1
y>1

x = 2^a [a is the length of x]
y = 2^b

x+3y<1000
2^a + 3(2)^b <1000

highest value of 2^a that can fit into this is 512

2^a = 512
a = 9

3(2)^b <1000-512
2^b < 488/3
2^b < 162
highest value of 2^b can be 128
b = 7

Maximum sum of length of x and length of y = a + b =9+7 = 16


Option D