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My favorite numbers (Posted on 2005-07-22) Difficulty: 3 of 5
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.

No Solution Yet Submitted by Jurgen    
Rating: 3.2000 (5 votes)

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Some Thoughts No Subject | Comment 3 of 13 |

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*D-K) = 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) +- (C-D)}/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 2005-07-22 03:51:26
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