Determine all possible sextuplets (A, B, C, D, E, F) of positive integers, with A ≤ B ≤ C, and, D ≤ E ≤ F and, A ≤ D, that satisfy both the equations: A+B+C = D*E*F and, A*B*C = D+E+F.

Prove that these are the only sextuplets that exist.

I too have the same result as  Jim (my listing is below) and like him I cannot give an explanation.

 1  1  6  2  2  2  -- 8  8  6  6
 1  1  7  1  3  3  -- 9  9  7  7
 1  1  8  1  2  5  --10  10  8  8
 1  2  3  1  2  3  -- 6  6  6  6
 1  2  5  1  1  8  -- 8  8  10  10
 1  3  3  1  1  7  -- 7  7  9  9

The Sextuples are to the left and the quads are the sum-product product-sum pairings of the equations


CLS
OPEN "Sum_Prod.txt" FOR OUTPUT AS #1
FOR a = 1 TO 30
FOR b = a TO 30
FOR c = b TO 30
FOR d = a TO 30
FOR e = d TO 30
FOR f = e TO 30
j = a + b + c
k = a * b * c
l = d + e + f
m = d * e * f
IF (j = m AND l = k) THEN
PRINT a; b; c; d; e; f; j; m; k; l
PRINT #1, a; b; c; d; e; f; j; m; k; l
END IF
NEXT: NEXT: NEXT: NEXT: NEXT: NEXT
CLOSE 1

Posted by brianjn on 2009-12-21 19:32:54