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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....
How is 2222 congruent to 3 mod 7 and 5555 congruent to 4 mod 7? What does congruent mean in this context ? I am totally lost ....Ian Stewart wrote: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.
As I said above, you don't need any of this for the GMAT. 'Congruent to' means 'has the same remainder as', and when you learn 'modular arithmetic', you learn how to do arithmetic with remainders -that's all I was doing above. Still, if you're preparing for the GMAT, you don't need to worry about any of this.acecoolan wrote: How is 2222 congruent to 3 mod 7 and 5555 congruent to 4 mod 7? What does congruent mean in this context ? I am totally lost ....
