Modular Arithmetic with Indices

Question

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

Exponentiation by squaring is what you want. – Nick Jan 07 '14 at 19:05 — Nick, Jan 07 '14 at 19:05

score 2 · Accepted Answer · answered Jan 07 '14 at 19:25

2

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

answered Jan 07 '14 at 19:25

Bill Dubuque

272,048

Could you please do it out so I can see how it is done I would greatly appreciate it – dunika Jan 07 '14 at 19:36
@user1552404 It is done. Put $,n=11,$ for your special case. – Bill Dubuque Jan 07 '14 at 19:37
Oh ye I see it now ha okay thanks – dunika Jan 07 '14 at 19:42

Modular Arithmetic with Indices

1 Answers1