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
quadruplets and proof
