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# number theory

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89
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What is the remainder when 2013^13 is divided by 13?

Jul 4, 2021

#1
+795
-1

$\frac{2013^{13}}{13} = 2013^{12}$

Remainder is trivial

#2
+26213
+2

What is the remainder when $$2013^{13}$$ is divided by $$13$$?

$$\begin{array}{|rcll|} \hline && \mathbf{2013^{13} \pmod{13}} \quad &| \quad 2013 \equiv 11 \pmod{13} \\ &\equiv&11^{13}\pmod{13} \quad &| \quad 11^4 \equiv 3 \pmod{13} \\ &\equiv&\left( 11\right)^{4*3+1} \pmod{13} \\ &\equiv& \left( 11^4 \right)^3*11 \pmod{13} \\ &\equiv& \left( 3 \right)^3*11 \pmod{13} \\ &\equiv& 27*11 \pmod{13}\quad &| \quad 27 \equiv 1 \pmod{13} \\ &\equiv& 1*11 \pmod{13} \\ &\equiv& \mathbf{ 11 \pmod{13} } \\ \hline \end{array}$$

The reminder is 11

Jul 4, 2021