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# number theory help

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Calculate $40^{13} \pmod{85}.$ Express your answer as a non-negative integer that is less than 85.

Jul 28, 2021

#1
+35345
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40^13 mod 85 = 10

Jul 28, 2021
#2
+26228
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Calculate $$40^{13} \pmod{85}$$.
Express your answer as a non-negative integer that is less than $$85$$.

$$\begin{array}{|rcll|} \hline 40^{3} \pmod{85} &\equiv& 80 \pmod{85} \\ &\equiv& 80-85 \pmod{85} \\ &\equiv& -5 \pmod{85} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline 40^{13} \pmod{85} &\equiv& 40^{3*4+1} \pmod{85} \\ &\equiv& 40^{3*4}*40 \pmod{85} \\ &\equiv& \left(40^3\right)^4*40 \pmod{85} \\ &\equiv& \left(-5\right)^4*40 \pmod{85} \\ &\equiv& 625*40 \pmod{85} \\ &\equiv& 25000 \pmod{85} \\ \mathbf{40^{13} \pmod{85}} &\equiv& \mathbf{ {\color{red}10} \pmod{85}} \\ \hline \end{array}$$

Jul 30, 2021