Least Positive Residue using Fermat's Litl Thm

**BlueAngel** · Mar 22nd 2009, 12:14 PM

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?

**ThePerfectHacker** · Mar 22nd 2009, 07:40 PM

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 $\pi \cdot 10^{24}$ is abusive but whatever I understand. Also, what you write above is not correct if you check with a calculator.

Let $N = 16q+r$ by the division algorithm. Then $2^N = 2^{16q+r} = \left( 2^{16}\right)^q \cdot 2^r \equiv 2^r(\bmod 17)$

This is what you ought to do.

**BlueAngel** · Mar 24th 2009, 09:38 AM

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?

**BlueAngel** · Mar 24th 2009, 10:51 PM

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?

**ThePerfectHacker** · Mar 25th 2009, 06:47 PM

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

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