Use Fermat’s little theorem to prove that $2222^{5555}+5555^{2222}$ is a multiple of $7$.
First, by using Fermat’s little theorem we have $$5555^{2222}=5555^{6\times 370+2}\equiv 5555^2\pmod 7$$ and $$2222^{5555}=2222^{6\times 925+5}\equiv 2222^5\pmod 7.$$ Therefore $$5555^{2222}+2222^{5555}\equiv 5555^2+2222^5\pmod 7$$ And I’m stuck from here.
