Ian Stewart wrote:

sudhir3127 wrote:Find the remainder of

(2222^5555 + 5555^7777)/7

Answer will follow soon....

Well, this isn't a real GMAT question. I'd use modular arithmetic, which you don't need for the GMAT. By ~, I mean 'congruent to' (if anyone reading this doesn't know what that means, don't worry about it!).

2222 ~ 3 mod 7
2222^5555 ~ 3^5555 mod 7
3^5555 = (3^2)*(3^5553) = (3^2)*[(3^3)^1851]

Since 3^3 = 27 ~ -1 mod 7, we have

(3^2)*[(3^3)^1851] ~ 2*(-1)^1851 mod 7 ~ -2 mod 7 ~ 5 mod 7.

So the remainder is 5 when 2222^5555 is divided by 7.

Similarly for 5555^7777:

5555 ~ 4 mod 7
4^7777 = 4 * 4^7776 = 4 * (4^3)^2592 ~ 4 * 1^2592 mod 7 ~ 4 mod 7.

(here, using that 4^3 ~ 1 mod 7).

So

(2222^5555 + 5555^7777) ~ 4+ 5 mod 7 ~ 2 mod 7.

The remainder should be 2, unless I've made an arithmetic mistake, which wouldn't surprise me, considering how quickly I'm doing this.