2021 Fall AMC 10B Problem 8

The greatest prime number that is a divisor of 16{,}384 is 2 because 16{,}384 = 2^{14}. What is the sum of the digits of the greatest prime number that is a divisor of 16{,}383?

\textbf{(A)} \: 3\qquad\textbf{(B)} \: 7\qquad\textbf{(C)} \: 10\qquad\textbf{(D)} \: 16\qquad\textbf{(E)} \: 22

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