(In reply to re: solution
by Steve Herman)
DM, Steve is correct! Here's why:
From (ii), 576 (1/B + 1/C) = 120 - 576 (1/A + 1/D)
i.e., 576 (B + C)/BC = 120 - (A + D), since AD = 576
The RHS of this equation is a positive integer, so BC must also be a factor of 576. From the list of possible factors, we conclude that only (A, B, C, D) = (8, 24, 36, 72) and (12, 16, 24, 28) are feasible solutions.
Great problem. KS!