Solve for x when $2222^{5555} + 5555^{2222} \equiv x \pmod{7}$

Question

I need to find the remainder when $2222^{5555} + 5555^{2222}$ is divided by $7$. I'm thinking that Fermat's Little Theorem might help. Any suggestions?

Answer 1

We see that $2222^{5555} \equiv_7 3^{5555}$. We have that $3^6 \equiv_7 1$, so $3^{5555} \equiv_7 3^5$. Similarly, $5555^{2222} \equiv_7 4^{2222} \equiv_7 4^2$. Together, we get $2222^{5555} + 5555^{2222} \equiv_7 3^5 + 4^2 \equiv_7 243 + 16 \equiv_7 0$