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Divide The Cubic, Get The Pairs (Posted on 2007-05-26) Difficulty: 3 of 5
Find all possible positive integers (p,q) with p≤q such that (q³+1)/(pq-1) is an integer.

  Submitted by K Sengupta    
Rating: 4.0000 (2 votes)
Solution: (Hide)
(p, q) = (1, 2), (1, 3), (2, 1), (2, 2), (2, 5), (3, 1), (3, 5), (5, 2), (5, 3).

Applying the restriction: p≤ q, we have:
(p,q) = (1.2), (1,3), (2,2), (2,3), (3,5)

EXPLANATION: Refer to the solution submitted by Brian Smith in this location.

(p,q) = (1, 2), (1, 3), (2, 1), (2, 2), (2, 5), (3, 1), (3, 5), (5, 2), (5, 3)

In consonance with the Solution to Problem B1 of 35th IMO 1994, the possible solutions without the restriction p<=q are given by:

(p,q) = (1, 2), (1, 3), (2, 1), (2, 2), (2, 5), (3, 1), (3, 5), (5, 2), (5, 3).

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re(2): Solution (a few questions)Brian Smith2007-05-28 23:59:42
re: Solution (a few questions)ken2007-05-27 20:12:01
re(2): Solution (spoiler)Brian Smith2007-05-27 14:32:40
re: Solution (spoiler)Dej Mar2007-05-27 02:25:59
SolutionSolutionBrian Smith2007-05-27 00:56:17
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