 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  Three Product Tease (Posted on 2014-06-12) 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 No Rating Comments: ( Back to comment list | You must be logged in to post comments.) 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 Please log in:

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