2024 AMC 12A Problem 11

There are exactly K positive integers b with 5 \leq b \leq 2024 such that the base-b integer 2024_b is divisible by 16 (where 16 is in base ten). What is the sum of the digits of K?

\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }20\qquad\textbf{(E) }21

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