2007 Cayley Problem 24

The number 8 is the sum and product of the numbers in the collection of four positive integers \{1,1,2,4\}, since 1+1+2+4=8 and 1 \times 1 \times 2 \times 4 = 8. The number 2007 can be made up from a collection of n positive integers that multiply to 2007 and add to 2007. What is the smallest value of n with n > 1?

\textbf{(A)}\ 1171\quad \textbf{(B)}\ 1337\quad \textbf{(C)}\ 1551\quad \textbf{(D)}\ 1777\quad \textbf{(E)}\ 1781

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.