Method to find solution for $a^x \equiv \mod n$ The congruence $5^x \equiv 1 \mod 36$ has a solution because $5$ and $36$ are relatively prime, i.e. $5$ and $2^23^2$ have no common factors.
Is there a method to find $x$? 
All I can see is that $5^3 \equiv 5 \mod 6$.

You could always use the Chinese Remainder Theorem for congruences like this.
For
$$
5^x\equiv 1 \ \bmod 36
$$
you should solve
$$
\begin{cases}
5^x\equiv 1 \ \bmod 4\\
5^x\equiv 1 \ \bmod 9
\end{cases}
$$
This way you have to deal with smaller moduli.
