2016 AMC 10A Problem 23

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 $p+q?$