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Three Product Tease (Posted on 2014-06-12) Difficulty: 3 of 5
Define P(n) as the sum of all the positive integers that are less than n and relatively prime to n.

Find all possible positive integer values of n such that P(n) = 3n, and prove that there are no others.

No Solution Yet Submitted by K Sengupta    
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Solution Sketch of proof Comment 2 of 2 |

I'm not sure this works but I think it does:

For any n, if k < n is relatively prime to n, then so is n-k. 

Therefore for any n, you can create pairs of numbers (k, n-k) which are both relatively prime to n and obviously sum to n.

Then the only way P(n) = 3n is where phi(n) is 6.  This is the case for n = 7, 9, 14, and 18. 

The last step would be proving that phi(n) != 6 for all other integers.

  Posted by tomarken on 2014-06-12 12:29:35
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