Question

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 < 1,000, what is the maximum possible sum of the length of x and the length of y?

Answer 1

16

Explanation:

We must find integers x, y with the most amount of prime divisors, not necessarily distinct, such that x + 3y < 1,000.

Obviously, this is achieved when the divisor is the least prime 2. So, we must find integers n, m such that

`\large 2^n + 3*2^m < 1,000`

since
`\large 2^10 = 1,024` , then n must be 9. For n=9 we find the greatest integer m such that

`\large 2^9 + 3*2^m <1,000`

and we find m=7

and
`\large x=2^9` ,
`\large y=2^7` are the numbers we are looking for and the sum of their length is 9+7 = 16.

So, 16 is the maximum possible sum of the length of x and the length of y.