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What is the remainder when 5^(301) is divided by 7?

 May 16, 2021
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What is the remainder when 5^(301) is divided by 7?

 

\(5^{301}\quad \mod7\\ =5*5^{300}\quad \mod7\\ =5*{(5^{3})}^{100} \quad \mod7\\ =5*{125}^{100} \quad \mod7\\ \equiv 5*{6}^{100} \quad \mod7\\ \equiv 5*{(-1)}^{100} \quad \mod7\\ \equiv 5*1 \quad \mod7\\ \equiv 5 \quad \mod7\\\)

 

 

 

 

 

LaTex:

5^{301}\quad \mod7\\
=5*5^{300}\quad \mod7\\
=5*{(5^{3})}^{100} \quad \mod7\\
=5*{125}^{100} \quad \mod7\\
\equiv 5*{6}^{100} \quad \mod7\\
\equiv 5*{(-1)}^{100} \quad \mod7\\
\equiv 5*1 \quad \mod7\\
\equiv 5 \quad \mod7\\

 May 17, 2021

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