AMC 12 2016a Problem 20

A binary operation $\diamondsuit$ has the properties that $a\ \diamondsuit\ (b\ \diamondsuit\ c) = (a\ \diamondsuit\ b)\cdot c$ and that $a\ \diamondsuit\ a = 1$ for all nonzero real numbers $a, b$ and $c.$ (Here the dot $\cdot$ represents the usual multiplication operation.) The solution to the equation $2016\ \diamondsuit\ (6\ \diamondsuit\ x) = 100$ can be written as $\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. What is $p + q?$

\textbf{(A)}\ 109 \textbf{(B)}\ 201 \textbf{(C)}\ 301 \textbf{(D)}\ 3049 \textbf{(E)}\ 33,601