Can you easily simplify large exponents without Fermat's Little Theorem?

Question

I am asked to check if $x = 19$ is a solution to the following congruence:

$$ x^{30034} ≡ 2 \pmod{18}$$

How can I do this? And in general, is there an easy/fast way to solve these types of problems when you can't use Fermat's LT?

Answer 1

Well, $19\equiv 1\pmod{18}$, so it clearly is not a root of this equation.

You can show that this equation has no solutions since $y^2\equiv 2\pmod{3}$ has no solution, and $x^{30034}$ is a perfect square.