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, 8 months ago

Log in to reply

@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, 7 months ago

is read as "12 congruent to 1 mod 11" or "12 equivalent to 1 mod 11"
–
Aneesh Kundu
·
2 years, 7 months ago

Log in to reply

@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, 7 months ago

@Aneesh Kundu
–
please explain it or give another way please please.........
–
Gautam Sharma
·
2 years, 8 months ago

Log in to reply

@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, 8 months ago

Log in to reply

@Aneesh Kundu
–
NO Problem. BTW IT WAS YOUR QUESTION. I got k=3 but was unable to get remainder.
–
Gautam Sharma
·
2 years, 8 months ago

## Comments

Sort by:

TopNewest\(\phi(2014)=936\)

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, 8 months ago

Log in to reply

– Gautam Sharma · 2 years, 7 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?Log in to reply

is read as "12 congruent to 1 mod 11" or "12 equivalent to 1 mod 11" – Aneesh Kundu · 2 years, 7 months ago

Log in to reply

– Marc Vince Casimiro · 2 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)Log in to reply

here – Aneesh Kundu · 2 years, 7 months ago

U may find itLog in to reply

– Gautam Sharma · 2 years, 8 months ago

please explain it or give another way please please.........Log in to reply

– Aneesh Kundu · 2 years, 8 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.Log in to reply

– Gautam Sharma · 2 years, 8 months ago

NO Problem. BTW IT WAS YOUR QUESTION. I got k=3 but was unable to get remainder.Log in to reply

@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, 8 months ago

Log in to reply

– Gautam Sharma · 2 years, 8 months ago

Thanks D);.Log in to reply