Thread: Least Positive Residue using Fermat's Litl Thm

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?

Originally Posted by BlueAngel

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?

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?

Originally Posted by BlueAngel

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?

You can reduce $\displaystyle 2^N$ to $\displaystyle 2^r$ where $\displaystyle N$ is that large number. Once you have $\displaystyle 2^r$ just reduce this number (which is not so large) mod $\displaystyle 17$.