# Modular Arithmetic with Indices

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

Hint $\rm\ mod\ 13\!:\ \color{#c00}{10\equiv -3} \Rightarrow \color{#c00}{10}^{3n}\equiv (\color{#c00}{-3})^{3n}\equiv (-27)^n\equiv (-1-2\cdot 13)^n\equiv (-1)^n$
• @user1552404 It is done. Put $\,n=11\,$ for your special case. – Bill Dubuque Jan 7 '14 at 19:37