Can anyone tell me how find the least positive residue of
2^(Pi)10^24(mod17)
or
2^314592653589793238462643383279502884197169399375 10020(mod17)
using Fermat's Little theorem?
Writing $\displaystyle \pi \cdot 10^{24}$ is abusive but whatever I understand. Also, what you write above is not correct if you check with a calculator.
Let $\displaystyle N = 16q+r$ by the division algorithm. Then $\displaystyle 2^N = 2^{16q+r} = \left( 2^{16}\right)^q \cdot 2^r \equiv 2^r(\bmod 17)$
This is what you ought to do.
Thanks for the response! im new to this forum so I cant quite figure out how to put all the symbols and stuff into place!
I still don't get it tho...I mean I get Fermat's Little Theorem, but I dont get how this helps solve for the Least Positive Residue?