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
(3^5)^1111 + (4^2)^1111
on dividing above number by 7 we get remainder as
5^1111 + 2^1111 ----------------------- Eqn 1
Since divisor is 7, remainder 5 is same as remainder -2. This will simplify calculations a lot. For simpleton souls I have also solved it taking Eqn 1. Those who are comfortable with above step, the riddle is almost solved, as remainder now becomes
= (-2)^1111 + 2^1111
= -(2^1111) + 2^1111
= 0
Thus the given number is divisible by 7.
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From Eqn 1, Remainder is : 5^1111 + 2^1111
5^1111 => (((5^3)*(5^3))^185)*5 => (36^185)*5 => 1*5 = 5
2^1111 => (8^370)*2 => 1*2 = 2
Thus remiander becomes 5 + 2 = 7 = 0