Exponential remainders?

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

#Algebra #NumberTheory #Remainder-FactorTheorem #Division

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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Comments

$\phi(2014)=936$

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 - 6 years, 7 months ago

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

Gautam Sharma - 6 years, 7 months ago

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 - 6 years, 7 months ago

@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 - 6 years, 7 months ago

@Calvin Lin – Thanks D);.

Gautam Sharma - 6 years, 7 months ago

@Aneesh Kundu – NO Problem. BTW IT WAS YOUR QUESTION. I got k=3 but was unable to get remainder.

Gautam Sharma - 6 years, 7 months ago

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 - 6 years, 6 months ago

U may find it here

Aneesh Kundu - 6 years, 6 months ago

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 - 6 years, 6 months ago

$12\equiv 1 \mod 11$

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

Aneesh Kundu - 6 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$