What is the largest integer $n$ that satisfies $(100^2-99^2)(99^2-98^2)\dots(3^2-2^2)(2^2-1^2)$ is divisible by $3^n$?

$ 
\textbf{(A)}\ 49
\qquad\textbf{(B)}\ 53
\qquad\textbf{(C)}\ 97
\qquad\textbf{(D)}\ 103
\qquad\textbf{(E)}\ \text{None of the above}
$