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How would I find $x$ where $\displaystyle 10^{33} \equiv x (\mod 13)$?

I have an exam coming up and I'm not sure how to do this. I am assuming this can be done without a calculator but if not could someone tell me otherwise? I'd really appreciate any help

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Hint $\rm\ mod\ 13\!:\ \color{#c00}{10\equiv -3} \Rightarrow \color{#c00}{10}^{3n}\equiv (\color{#c00}{-3})^{3n}\equiv (-27)^n\equiv (-1-2\cdot 13)^n\equiv (-1)^n$

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  • $\begingroup$ Could you please do it out so I can see how it is done I would greatly appreciate it $\endgroup$ – dunika Jan 7 '14 at 19:36
  • $\begingroup$ @user1552404 It is done. Put $\,n=11\,$ for your special case. $\endgroup$ – Bill Dubuque Jan 7 '14 at 19:37
  • $\begingroup$ Oh ye I see it now ha okay thanks $\endgroup$ – dunika Jan 7 '14 at 19:42

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