61.0k views
5 votes
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?

1 Answer

2 votes

Answer:

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.

User Dthrasher
by
7.9k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories