2014 Pascal Problem 25

Suppose that \dfrac{2009}{2014} + \dfrac{2019}{n} = \dfrac{a}{b}, where a, b and n are positive integers with \dfrac{a}{b} in lowest terms. What is the sum of the digits of the smallest positive integer n for which a is a multiple of 1004?

\textbf{(A)}\ 16\quad \textbf{(B)}\ 17\quad \textbf{(C)}\ 14\quad \textbf{(D)}\ 20\quad \textbf{(E)}\ 21

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 Pascal contests here.