How to solve this modular equation? $x^{19} \equiv 36 \mod 97$. How to solve the following? $x^{19} \equiv 36 \mod 97$.
I am having trouble figuring this out. Which technique do I need to use? Chinese Remainder or Fermat's Little Theorem? 

Note that $x^{96}\equiv 1 \mod 97$ and since $95=5\times 19$, we have $$x^{96}=x(x^{19})^5\equiv x(36)^5\equiv 1$$
This you should be able to solve using elementary means (e.g. by computing an inverse of $36 \mod 97$)
