Prove that ` 2222^(5555) + 5555^(2222) ` is
divisible by 7 .

Explanation

We have ` , 2222^(5555) + 5555^(2222) `
`= (2222^(5555) + 4^(5555))+ (5555^(2222) - 4^(2222) - (4^(5555) - 4^(2222) ) ` …(i)
The number ` (2222^(5555) + 4^(5555) )` is divisible by
` 2222 + 4 = 2226 = 7 xx318` , which is divisible by 7 and the number
` (5555^(2222) - 4 ^(2222) )` is divisaible by
` 5555- 4 = 5551 = 7xx793 ` , which is diviisible by 7 and the number
` (4^(5555) - 4^(2222)) = 4^(2222) (4^(3333) -1) = 4^(2222)(64^(1111) - 1^(1111)) ` is
divisible by ` 64 - 1 = 63 = 7xx9 ` , which is divisible by 7.
` 2222^(5555) + 5555^(2222)` is divisible by 7