Find all possible positive integers A, B and C - with A ≤ B, that simultaneously satisfy:

- A+B-C=12, and:
- A
^{2} + B^{2} - C^{2} = 12

(1) C = A+B-12

(2) Substituting in the 2nd equation and simplifying gives

-156 = 24A + 24B - 2AB

(3) Solving for B gives B = (-78-12A)/(12-A)

(4) Let D = 12-A. Then B = 12 - 66/D

(5) The only possible values for D are +/- 1,2,3,6,11,22,33,66.

So there are 16 possible solutions.

They are:

D B A C

1 -54 11 -55

2 -21 10 -23

3 -10 9 -13

6 1 6 -5

11 6 1 -5

22 9 -10 -13

33 10 -21 -23

66 11 -54 -55

-1 78 13 79

-2 45 14 47

-3 34 15 37

-6 23 18 29

-11 18 23 29

-22 15 34 37

-33 14 45 47

-66 13 78 79

**The only solutions with A, B and C positive and A <= B are**

(13,78,79), (14,45,47), (15,34,37) and (18,23,29)