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Math Help - Mod and primes?!

  1. #1
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    Mod and primes?!

    Is it true that

    1.if p is prime then n^(p squared)=n mod p for all natural numbers n??

    2. n^7=n mod 35 for all natural numbers n

    If so then what working should i show?and if not then what would be a counter-example??thanksss~~~
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by bryan06 View Post
    2. n^7=n mod 35 for all natural numbers n
    n = 2 is a counterexample.

    -Dan
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by bryan06 View Post
    1.if p is prime then n^(p squared)=n mod p for all natural numbers n??
    This one is true by a version of Fermat's Little Theorem. We can state that


    since p^2 - 1 is divisible by p - 1. So by Fermat's Little


    -Dan
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  4. #4
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    thankss~~still a bit confused though i researched on wikipedia what fermat little theorem is but still cant figure out how you derived the following:
    Quote Originally Posted by topsquark View Post
    also how do you get from this line
    Quote Originally Posted by topsquark View Post
    since p^2 - 1 is divisible by p - 1.
    to this line

    Quote Originally Posted by topsquark View Post
    So by Fermat's Little

    thankss
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  5. #5
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by topsquark View Post
    This one is true by a version of Fermat's Little Theorem. We can state that


    since p^2 - 1 is divisible by p - 1. So by Fermat's Little


    -Dan
    Or just note that
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