Power cycles modulo $n$

Powers.
- $2^0=1$
- $2^1=2$
- $2^2=4$
- $2^3=8\equiv1$
- $2^4\equiv2$
- $2^5\equiv4$
- ...

Period. $3$, generated by $2^3\equiv1$.

Conclusion. The sequence is periodic with period $3$: $1,2,4,1,2,4,\ldots$.

General. The period of $a\bmod n$ is $\operatorname{ord}_n(a)$, which divides $\varphi(n)$.