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.

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**