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My favorite numbers IV (Posted on 2010-05-08) Difficulty: 3 of 5
Determine all possible quadruplet(s) (A, B, C, D) of positive integers, with A ≥ C, that satisfy this system of equations:

A/B = C-D, and:

C/D = A-B

Prove that these are the only quadruplet(s) that exist.

No Solution Yet Submitted by K Sengupta    
Rating: 1.0000 (1 votes)

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Solution quadruplets and proof Comment 3 of 3 |
Here's a proof, which leads to three quadruplets that satisfy the equations:

Multiplying both equations by D gives 

(1) A = BC - BD, and
(2) C = AD - BD

Sustituting (1) into (2) gives:

(3) C = BCD - BDD - BD

Solving (3) for C gives:

(4) C = BD(D+1)/(BD-1)

Since C is integral, (BD-1) must divide BD(D+1) evenly. 

If (BD-1) <> 1, then it is relatively prime to BD and must divide (D+1) evenly.
And if (BD-1) = 1, then it also divides (D+1) evenly.

(5) Therefore, (BD-1) divides (D+1) evenly, so necessarily

    (BD-1) <= (D + 1), which solves to 

(6) (B-1)D <=2
    which means that (B,D) = (1,1) or (1,2) or (2,1) or (2,2) or (3,1)    

This leads to the following:

B  D  C  A
1  1   X     Does not give valid C
1  2   6  4  Not a solution: C > A
2  1   4  6  Solution 
2  2   4  4  Solution
3  1   3  6  Solution

So, there are only three solutions:

(4,2,4,2) and (6,2,4,1) and (6,3,3,1)

Edited on May 9, 2010, 11:13 am
  Posted by Steve Herman on 2010-05-09 11:10:14

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