Find $144^3$ mod $213$

Question

Find $144^3$ mod $213$

I'm not sure how to solve this.

I know that $213=3\times 71$, which are primes.

And I can find that $144\equiv 0$ mod $3$, and $144\equiv 2$ mod $71$.

Answer 1

$144^3\equiv0\pmod3$ and $144^3\equiv2^3=8\pmod{71}$.

If $x\equiv8\pmod{71},$ then $x\equiv79, 150, $ or $221\pmod{213}$.

Can you take it from here?

Answer 2

$144^{\large 3}\bmod\, 3\cdot 71\, =\, 3\:\!\underbrace{\left[\dfrac{\color{#0a0}{144^{\large 3}}}3\!\bmod 71\right]\!=\, 3\:\![\color{#c00}{50}]}_{\textstyle \ \dfrac{\color{#0a0}{2^{\large 3}}}3\,\equiv\, \dfrac{-63}3\,\equiv\,\color{#c00}{-21}}$