2004 AMC 12B 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 \qquad\mathrm{(B)}\ 250,250 \qquad\mathrm{(C)}\ 250,500 \qquad\mathrm{(D)}\ 250,750 \qquad\mathrm{(E)}\ 251,000

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.