+0

-1
165
3

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

Jun 28, 2021

#1
+238
+1

The tens digit is 3

Jun 28, 2021
#2
0

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

The tens digit is 9

Jun 28, 2021
#3
+26228
+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 $${\color{red}3}$$

Jun 28, 2021
edited by heureka  Jun 28, 2021