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.

(In reply to

As simple as possible, but no simpler... by ed bottemiller)

Format problem on the last posting

1+1+1+9 = 12 = 1 * 1 *3 * 4

1 * 1 * 1 * 9 = 9 = 1+1+3+4

Or, if "integers" need not be single digits:

e.g. (1,1,1,11) and (1,1,2,7)

1+1+1+11 = 14 = 1 * 1 * 2 * 7

1 * 1 * 1 * 11 = 11 = 1+1+2+7

and others.