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sudhir3127
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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!).sudhir3127 wrote:Find the remainder of
(2222^5555 + 5555^7777)/7
Answer will follow soon....
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.
