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What is the tens digit of $17^{1993}?$

 Jun 28, 2021
 #1
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+1

The tens digit is 3

 Jun 28, 2021
 #2
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0

According to WolframAlpha: https://www.wolframalpha.com/input/?i=17%5E1993

 

The tens digit is 9

 Jun 28, 2021
 #3
avatar+26213 
+2

What is the tens digit of \(17^{1993}\)?

 

\(\small{ \begin{array}{|rcll|} \hline \mathbf{17^{1993} \pmod{100}} &\equiv& \left( 17^2 \right)^{996}+1 \pmod{100} \\ &\equiv& \left( 17^2 \right)^{996}*17 \pmod{100} \\ && \boxed{17^2 = 289 \equiv 89 \equiv 89-100 \equiv -11 \pmod{100}} \\ &\equiv& \left( -11 \right)^{996}*17 \pmod{100} \\ &\equiv& \left( (-11)^2 \right)^{498}*17 \pmod{100} \\ &\equiv& \left( 121 \right)^{498}*17 \pmod{100} \quad | \quad 121 \equiv 21 \pmod{100} \\ &\equiv& \left( 21 \right)^{498}*17 \pmod{100} \\ &\equiv& \left( 21 \right)^{5*99+3}*17 \pmod{100} \\ &\equiv& \left( 21^5 \right)^{99}*21^3*17 \pmod{100} \quad | \quad 21^5 \equiv 01 \pmod {100} \\ &\equiv& \left( 1\right)^{99}*21^3*17 \pmod{100} \\ &\equiv& 21^3*17 \pmod{100} \\ &\equiv& 157437 \pmod{100} \\ &\equiv& \mathbf{{\color{red}3}7 \pmod{100}} \\ \hline \end{array} }\)

 

The tens digit is \({\color{red}3}\)

 

laugh

 Jun 28, 2021
edited by heureka  Jun 28, 2021

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