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What is the remainder when 2013^(2013^2) is divided by 13?

 May 27, 2021
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Notice that $2013\equiv 11\equiv -2\pmod{13}$.

 

So $2013^{2013^2}\equiv (-2)^{2013^2}\equiv -2^{2013^2}\pmod{13}$ since $2013^2$ is odd.

 

Notice that $\gcd(2,13)=1$ and $13$ is prime, so we can invoke fermat's little theorem, i.e. $2^{12}\equiv 1\pmod{13}$.

 

Notice that $2013\equiv 9\pmod{12}\implies 2013^2\equiv 9^2=81\equiv 9\pmod{12}$.

 

Hence $-2^{2013^2}\equiv -2^9\pmod{13}$. Note $2^9\equiv 512\equiv 5\pmod{13}$, hence the answer is $-5\equiv \boxed{8}\pmod{13}$.

 May 27, 2021

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