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Composite Integer Conclusion (Posted on 2016-01-18) Difficulty: 3 of 5
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

No Solution Yet Submitted by K Sengupta    
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Solution 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
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