Find 4 different positive integers A,B,C,D for which:
A+B = C*D and
A*B = C+D
How many sets of 4 numbers can you find? Prove that only those sets exist.
B = (C+D)/A
A + (C+D)/A = C*D
A^2  (C*D)*A + (C+D) = 0
A = {(C*D) + sqrt [ C^2*D^2  4*(C+D)]}/2
C^2*D^2  4*(C+D) = K^2
C^2*D^2  K^2 = 4*(C+D)
(C*D+K)*(C*DK) = 4*(C+D)
A = (C*D + K)/2 > A1 = (C*D + K)/2
A2 = (C*D  K)/2
4*A1*A2 = 4*(C+D) > A1*A2 = (C+D)
A1 + A2 = C*D
x^2  (C+D)x + C*D = 0
x = {(C+D) + sqrt[C^2 + 2*C*D + D^2  4*C*D)}/2
x = {(C+D) + (CD)}/2
x1 = C, and x2 = D.
The sets exist for A (or B) = C (or D).
Since are required different integers, the set is null.

Posted by pcbouhid
on 20050722 03:51:26 