1999 Cayley Problem 17

Let N be the smallest positive integer whose digits have a product of 2000. The sum of the digits of N is

\textbf{(A)}\ 21\quad \textbf{(B)}\ 23\quad \textbf{(C)}\ 25\quad \textbf{(D)}\ 27\quad \textbf{(E)}\ 29

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Cayley contests here.