AMC 12 2024a 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 $K$?

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