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'.

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TopNewest2^2015 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, 7 months ago

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– Adarsh Kumar · 2 years, 7 months ago

yes awesome!!Log in to reply

– Krishna Ar · 2 years, 7 months ago

Good. Nice solution using CRT. Adarsh KumarLog in to reply

@Calvin Lin ? – Adarsh Kumar · 2 years, 7 months ago

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@Satvik Golechha @Krishna Ar ? – Adarsh Kumar · 2 years, 7 months ago

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this problem right? – Krishna Ar · 2 years, 7 months ago

And it wasLog in to reply

– Adarsh Kumar · 2 years, 7 months ago

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.Log in to reply

– Krishna Ar · 2 years, 7 months ago

Answer's \(8\)Log in to reply

– Adarsh Kumar · 2 years, 7 months ago

Yes,I actually know the answer but could you please provide a step-by-step solution?Log in to reply

– Krishna Ar · 2 years, 7 months ago

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!!Log in to reply

– Adarsh Kumar · 2 years, 7 months ago

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.Log in to reply

– Krishna Ar · 2 years, 7 months ago

Yes, but it still provides the answer! ^_^. I can't think of any other method.Log in to reply

– Subrata Saha · 2 years, 7 months ago

You should always have CHINESE REMAINDER THEOREM in mind while solving this type of problemsLog in to reply

– Adarsh Kumar · 2 years, 7 months ago

well,thanx for your help but if you can think of any other method please post!!Log in to reply

– Krishna Ar · 2 years, 7 months ago

SureLog in to reply

– Adarsh Kumar · 2 years, 7 months ago

and I think that results are important but the method is more important.Log in to reply