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My favorite numbers III (Posted on 2010-03-04) Difficulty: 4 of 5
Determine all possible octuplets (A, B, C, D, E, F, G, H) of positive integers, with A ≤ B ≤ C ≤ D, and, E ≤ F ≤ G ≤ H and, A ≤ E, that satisfy both the equations: A+B+C+D = E*F*G*H and, A*B*C*D = E+F+G+H.

Prove that these are the only octuplets that exist.

No Solution Yet Submitted by K Sengupta    
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solution without proof Comment 6 of 6 |
There are five octuplets that satisfy both equations:
1.( 1, 1, 1, 9, 1, 1, 3, 4)
2.( 1, 1, 1,11, 1, 1, 2, 7)
3.( 1, 1, 2, 4, 1, 1, 2, 4)
4.( 1, 1, 2, 7, 1, 1, 1,11)
5.( 1, 1, 3, 4, 1, 1, 1, 9)

Two pairs of octuplets, (1 and 5) and (2 and 4), differ only in that A,B,C,D and E,F,G,H are swapped.

I'll leave the proof to the math professors.
  Posted by Dej Mar on 2010-03-04 21:51:56
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