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Exponential remainders?

can anyone help me in finding the remainder when \(3^{942}\) is divided by 2014?? I dont need answer just help!!!!! please.

Note by Gautam Sharma
2 years, 4 months ago

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\(\phi(2014)=936\)

And \(\gcd(3,2014)=1\)

By Euler-Fermat theorem we have \(3^{\phi(2014)}\equiv 3^{936}\equiv 1\mod 2014\)

\(\Rightarrow 3^{936}\times 3^6\equiv 3^{942}\equiv 729 \mod 2014\) Aneesh Kundu · 2 years, 4 months ago

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@Aneesh Kundu I want to ask How to read it or interpret this like when we do 12/3=4 , we say when 12 is divided by 3 we get 4 as a quotient.How to read it? Gautam Sharma · 2 years, 3 months ago

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@Gautam Sharma \(12\equiv 1 \mod 11\)

is read as "12 congruent to 1 mod 11" or "12 equivalent to 1 mod 11" Aneesh Kundu · 2 years, 3 months ago

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@Aneesh Kundu I always use Euler's totient function since it reduces exponents into a fathomable number. (Though I still do not know the proof of the theorem, would be great if someone presents one) Marc Vince Casimiro · 2 years, 3 months ago

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@Marc Vince Casimiro U may find it here Aneesh Kundu · 2 years, 3 months ago

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@Aneesh Kundu please explain it or give another way please please......... Gautam Sharma · 2 years, 4 months ago

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@Gautam Sharma Sorry for my mistake I've now edited it. There's no other method to my knowledge. U can look up this theorem in the brilliant wiki. Aneesh Kundu · 2 years, 4 months ago

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@Aneesh Kundu NO Problem. BTW IT WAS YOUR QUESTION. I got k=3 but was unable to get remainder. Gautam Sharma · 2 years, 4 months ago

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@Aneesh Kundu @GAUTAM SHARMA Check out Euler's Theorem in the Modular Arithmetic Wiki. That should provide you with explanations about how to approach problems like this. Calvin Lin Staff · 2 years, 4 months ago

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@Calvin Lin Thanks D);. Gautam Sharma · 2 years, 4 months ago

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