Is 2222⁵⁵⁵⁵ + 5555²²²² divisible by 7?

2222⁵⁵⁵⁵ + 5555²²²² = 3⁵⁵⁵⁵ + 4²²²² (mod 7). There are only 7 residue classes modulo 7. Therefore, the series of powers of a given number (modulo 7) is bound to be cyclic (of period 6). For example, 3² = 2 (mod 7), 3³ = 6 (mod 7), 3⁴ = 4 (mod 7), 3⁵ = 5 (mod 7), 3⁶ = 1 (mod 7), 3⁷ = 3 (mod 7), and so on: 3,2,6,4,5,1,3,... Note that 5555 = 5 (mod 6). Therefore, 2222⁵⁵⁵⁵ = 3⁵ (mod 7) = 5(mod 7).

Similarly, 5555²²²² = 4²²²² (mod 7). Powers of 4 modulo 7 form a cycle of period 3: 4,2,1,4,... Hence, 4²²²² (mod 7) = 4^{2222 ( mod 3)}(mod 7) = 4² (mod 7) = 2 (mod 7).

Adding up 5 + 2 gives a number divisible by 7. So is 2222⁵⁵⁵⁵ + 5555²²²².