Demonstrate that $2222^{5555}+5555^{2222}$ is divisible by $7$
2. We have $2222=7\cdot 317+3, \ 5555=7\cdot 793+4$
Then $(7k+3)^{5555}+(7l+4)^{2222}=7p+3^{5555}+4^{2222}=$
$=7p+243^{1111}+16^{1111}=7p+259q=7(p+37q)$