How to determine all possible values of $x$ to find congruency

Question

How would I find all the possible integer values of $x$ for the following congruence (or any of this form):

$3131x^{3131} + 2760x^{2761} \equiv 64$ (mod $93$)

Answer 1

Let's use a Chinese Remainder Theorem type of idea: Note that for $x$ to be a solution, $93$ must divide the left hand side which means that $3$ and $31$ also divide the left hand side. So, first, let's work $\mod{3}$ (and note that by FLT, $x^2\equiv 1\pmod{3}$ for $x\neq0$): $$3131x^{3131} + 2760x^{2761} \equiv 64\equiv x^{1}+0\equiv64\equiv1\implies x\equiv1\pmod{3}$$ Now, look at $\mod{31}$ (now, $x^{\phi(31)}=x^{30}\equiv1\pmod{31}$ for $x\neq0$): $$3131x^{3131} + 2760x^{2761} \equiv 64\equiv 0+1\cdot x^1\equiv 2\implies x\equiv 2\pmod{31}$$ Now, use the Chinese Remainder Theorem to lift up to the solutions $\pmod{3\cdot 31}$