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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
3 years, 1 month 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 - 3 years, 1 month ago

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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 - 3 years ago

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\(12\equiv 1 \mod 11\)

is read as "12 congruent to 1 mod 11" or "12 equivalent to 1 mod 11"

Aneesh Kundu - 3 years ago

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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 - 3 years ago

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U may find it here

Aneesh Kundu - 3 years ago

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please explain it or give another way please please.........

Gautam Sharma - 3 years, 1 month ago

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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 - 3 years, 1 month 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 - 3 years, 1 month 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 - 3 years, 1 month ago

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@Calvin Lin Thanks D);.

Gautam Sharma - 3 years, 1 month ago

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