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Multiply Together, Get Cubic (Posted on 2007-06-21) Difficulty: 4 of 5
Determine all possible integer pairs (p,q) such that p+q+s=pqs, where s=gcd(p,q) and gcd denotes the greatest common divisor.

No Solution Yet Submitted by K Sengupta    
Rating: 3.0000 (1 votes)

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Some Thoughts if s = 1 (one more solution!) | Comment 4 of 8 |
In all cases, p = (q^2 + s^3)/(qs - 1)

If p and q are relatively prime (s = 1), then

p = (q^2 + 1)/(q - 1) = (q - 1) + 2q/(q - 1)

P is therefore integral if q- 1 divides 2 (i.e, q = 3)
  or if q - 1 divides q (i.e, q = 2 or 0)

This leads to three solutions:
(5, 3), (5,2) and (-1,0)

I checked the definitiion, and the gcd of -1 and 0 is in fact 1!

Next step: s > 1

Edited on June 21, 2007, 7:01 pm
  Posted by Steve Herman on 2007-06-21 18:59:58

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