MathNet 2024-part-05 — 168 problems (EN–VI) Question 148

There are exactly $K$ positive integers $b$ with $5 \le b \le 2024$ 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 $K$?
(A) 16 (B) 17 (C) 18 (D) 20 (E) 21