2016 AMC 12A 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\qquad\textbf{(B)}\ 201\qquad\textbf{(C)}\ 301\qquad\textbf{(D)}\ 3049\qquad\textbf{(E)}\ 33,601

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