elementary number theory - Proof that $2^{222}-1$ is divisible by 3

How can I prove that $2^{222}-1$ is divisible by three?
I already have decomposed the following one: $(2^{111}-1)(2^{111}+1)$ and I understand I should just prove that $(2^{111}-1)$ is divisible by three or that $(2^{111}+1)$ is divisible by three. But how can I solve this problem?

Answer

The routine way is to invoke Fermat's little theorem:

$$a^{p-1}-1\equiv 0\,(\text{mod}\,p)$$ for $\mathrm{gcd}(a,p)=1$.
Plug in $a=2^{111},p=3$.