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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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Mutually inclusive | Comment 4 of 6 |
Like ed bottemiller I had some doubt as to if the equations were to be considered together or whether they could stand alone.

If the equations are to be considered together then:
 A,B,C,D are 1,1,2,4 as are E,F,G,H in the same order.

If the equations may stand alone then
   A  B  C  D  E  F  G  H
    1  1  1  9   1  1  3  4 or
this also holds true
    1  1  3  4   1  1  1  9

  Posted by brianjn on 2010-03-04 20:09:57
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