With the help of Chinese remainder theorem show that $ x^{144} \equiv 1 \pmod{ 323}$

Question

With the help of Chinese remainder theorem show that $ x^{144} \equiv 1 \pmod{323}$ for all $x$ relatively prime to 323.

The problem with me is that I used to use CRT when $x$ is raised to a power of 1, but how can I work with $x$ to the power of 144, could anyone explain this for me please?

Answer 1

We have $323=324-1=18^2-1=17\times 19$. So by CRT, it suffices to show $x^{144}\equiv 1\pmod{17}$ and $x^{144}\equiv 1\pmod{19}$ for all $x$ coprime to 17 and 19. Can you finish it from here?