AMC 12 2004b Problem 23

The polynomial $x^3 - 2004 x^2 + mx + n$ has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. How many values of $n$ are possible?

$\mathrm{(A)}\ 250,000 \mathrm{(B)}\ 250,250 \mathrm{(C)}\ 250,500 \mathrm{(D)}\ 250,750 \mathrm{(E)}\ 251,000$