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HELP!!!

Yesterday I encountered this problem while I was solving a problem.But don't worry the problem has been solved.Could somebody give a solution to this: \(2^{2015}\equiv a(mod{20}).\)You have to find 'a'.

Note by Adarsh Kumar
2 years, 3 months ago

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2^2015 is a multiple of 4 And 2^2015 leaves a remainder of 3 modulo 5 ( Fermat's little theorem) so, on dividing by 20, I.e.4*5, remainder is 8 as krishna said Subrata Saha · 2 years, 3 months ago

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@Subrata Saha yes awesome!! Adarsh Kumar · 2 years, 3 months ago

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@Subrata Saha Good. Nice solution using CRT. Adarsh Kumar Krishna Ar · 2 years, 3 months ago

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@Calvin Lin ? Adarsh Kumar · 2 years, 3 months ago

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@Satvik Golechha @Krishna Ar ? Adarsh Kumar · 2 years, 3 months ago

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@Adarsh Kumar And it was this problem right? Krishna Ar · 2 years, 3 months ago

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@Krishna Ar Yes,it was this problem but I figured out that the answer would not be 10 so I solved that problem but, I do not know how to find the remainder. Adarsh Kumar · 2 years, 3 months ago

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@Adarsh Kumar Answer's \(8\) Krishna Ar · 2 years, 3 months ago

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@Krishna Ar Yes,I actually know the answer but could you please provide a step-by-step solution? Adarsh Kumar · 2 years, 3 months ago

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@Adarsh Kumar Are you joking? (Really, coz u've solved my problem...)..ANyway...\( \phi(20)=8.\) thus, we have that \(2^{2015} \pmod {20} = 2^7 \pmod {20}\). Which is nothing but 8!! Krishna Ar · 2 years, 3 months ago

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@Krishna Ar I think you made a slight mistake as you have applied Euler's Totient theorem which can only be applied when the GCD(a,n)=1. Adarsh Kumar · 2 years, 3 months ago

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@Adarsh Kumar Yes, but it still provides the answer! ^_^. I can't think of any other method. Krishna Ar · 2 years, 3 months ago

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@Krishna Ar You should always have CHINESE REMAINDER THEOREM in mind while solving this type of problems Subrata Saha · 2 years, 3 months ago

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@Krishna Ar well,thanx for your help but if you can think of any other method please post!! Adarsh Kumar · 2 years, 3 months ago

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@Adarsh Kumar Sure Krishna Ar · 2 years, 3 months ago

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@Krishna Ar and I think that results are important but the method is more important. Adarsh Kumar · 2 years, 3 months ago

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