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Math Help - Least Positive Residue using Fermat's Litl Thm

  1. #1
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    Post 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?
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  2. #2
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    Quote Originally Posted by BlueAngel View Post
    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.
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    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?
    Last edited by BlueAngel; March 24th 2009 at 10:21 AM.
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    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?
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    Quote Originally Posted by BlueAngel View Post
    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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