2021 Fall AMC 10B Problem 14

Una rolls 6 standard 6-sided dice simultaneously and calculates the product of the 6{ } numbers obtained. What is the probability that the product is divisible by 4?

\textbf{(A)}\: \frac34\qquad\textbf{(B)} \: \frac{57}{64}\qquad\textbf{(C)} \: \frac{59}{64}\qquad\textbf{(D)} \: \frac{187}{192}\qquad\textbf{(E)} \: \frac{63}{64}

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