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Reciprocal Equation #5 (Posted on 2013-10-15) Difficulty: 3 of 5
Find all pairs (A, B) of distinct nonzero integers with A ≠ -1 and B ≠ -1 such that (1 + 1/A) is a nonzero integer multiple of (1 + 1/B).

Prove that there are no others.

Note: (1 + 1/A) is can be a negative as well as positive integer multiple of (1 + 1/B). So, equations like: (1 + 1/A) = -2(1 + 1/B) or, (1 + 1/A) = (1 + 1/B) are permissible. Remember, A and B must be distinct.

No Solution Yet Submitted by K Sengupta    
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Solution Rest of a solution Comment 3 of 3 |
(In reply to Part of a solution by Brian Smith)

Continuing from earlier.


Let C=-B then K = [C*(A+1)]/[A*(C-1)].  A and C are both positive integers.  Then for K to be an integer C>=A>=C-2.  This yields three cases:

Case 1: C=A
Then K = 1 + 2/(C-1).  For K to be an integer C must be 2 or 3.  This yields solutions (A,B,K) = (2,-2,3) and (3,-3,2).

Case 2: C-1=A
Then K = [C^2]/[(C-1)^2].  For K to be an integer C must be 2.  This yields a solution (A,B,K) = (1,-2,4).

Case 3: C-2=A
Then K = C/(C-2).  For K to be an integer C must be 3 or 4.  This yields solutions (A,B,K) = (1,-3,3) and (2,-4,2).

These five solutions are the only nontrivial integer solutions to K=(1+1/A)*(1+1/B).

Edited on July 4, 2016, 9:45 am
  Posted by Brian Smith on 2016-07-04 09:45:19

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