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.

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