What does $[5^{2000}]$ equal to in $\mathbb{Z}_4$?

Question

Question: What does $[5^{2000}]$ equal to in $\mathbb{Z}_4$?

My proof: (which I doubt whether its correct or not since it doesn't use the hint in the book)

$[5^{2000}]=([5])^{2000}$ Since $5 \equiv 1 (\mod 4)$, so $[5]=[1]$. So $[5^{2000}]=([1])^{2000}=[1]$.

Is my proof correct? Thanks!

Answer 1

It is perfectly correct. $$5 \equiv 1 \mod 4 \implies 5^{2000} \equiv 1^{2000} \mod 4 \implies 5^{2000} \equiv 1 \mod 4 \implies \left[ 5^{2000}\right]= [1].$$