Euclidean Algorithm with modular powers I have these two questions which go as:

1) Find $x \in \{0,1...16\}$ such that $x^{13}\equiv2\pmod{17}$

2) Find $x \in \{0,1...22\}$ such that $x^{13}\equiv3\pmod{23}$

I was wondering what the best way to approach these were, I was thinking using the Euclidean Algorithm?

This is what I have so far for 1):

We know hcf$((17-1),13)=1$, so $1=13s-16t$ for some s,t. But not sure how to go further? Maybe Fermat's Little Theroem but again not sure. Thank you!
Using Fermat's little theorem the first equation is equivalent to
$$x^{-3}\equiv2\mod{17}$$
$$2x^3\equiv1\mod{17}$$
$$x^3\equiv9\mod{17}$$
$$\therefore x\equiv-2\equiv15 \mod{17}$$
The second example seems like one can only solve with trial and error.
For the second example,
$$
16^{13}\equiv 3 \bmod 23.
$$
Also here we can compute $16^{11}\equiv 1\bmod 23$ as descibed in the comments, so that $16^{13}\equiv 16^2\equiv 3\bmod 23$.