perplexus.info :: Just Math : Composite Integer Conclusion

Composite Integer Conclusion (Posted on 2016-01-18)

Find all possible values of a composite positive integer N such that every divisor, D, of N (excluding N) satisfies:
N-20 ≤ D ≤ N-12

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)

Solution

Comment 2 of 2 |

I think this should have been D2.

For all D to satisfy the inequality, the square root of N must also satisfy it. Specifically N-20 <= sqrt(N).

Squaring each side and rearranging yields: N^2 - 41N + 400 <= 0. This means 16 <= N <= 25.

Manually checking each composite integer 16, 18, 20, 21, 22, 24, 25 finds N=21 (1 <= 3 and 7 <= 9) and N=25 (5 <= 5 <=13) are the only solutions.

Posted by Brian Smith on 2016-01-18 11:53:55

Please log in:

Forums (1)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (6)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:


perplexus dot info