2024 Mock AMC 12 - Problem 14

Suppose that d_1 is a divisor of 2024^4 such that if a second divisor d_2 \neq d_1 were chosen at random, there would be a \frac{1}{3} chance that d_1d_2 is a factor of 2024^4 as well. Given that the sum of all possible d_1 is S, how many factors does S have?

\textbf{(A) }18\qquad\textbf{(B) }36\qquad\textbf{(C) }48\qquad\textbf{(D) }96\qquad\textbf{(E) }150