Given number is divisible by 7.

This problem can be easily solved by focusing on the remainder when the given number is divided by 7.

Dividing (2222^5555 + 5555^2222) by 7 leaves following remainder

3^5555 + 4^2222 which is same as

THAN -----

 (3^2)[(3^3)^1851]  + (4^2)[(4^3)^740]

3^3 div by 7 gives remainder -1 , 4^3 div 7 gives rem +1

so we have 9 x (-1)^1851 + 16 x (1)^740 = 7 which wen divided by seven gives 0 as remainder