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Math Help - Euler's theorem

  1. #1
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    Euler's theorem

    Will anyone please show me a proof of Euler's theorem.

    Thank you soo much in advance!
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  2. #2
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    Quote Originally Posted by jenjen View Post
    Will anyone please show me a proof of Euler's theorem.
    Your last question was number theory related.
    I assume you want to show,
    a^phi(n) = 1 (mod n)

    I have two proofs.
    1)The elementary number theory proof.
    2)Group theory proof, it is shorter.

    Which one will it be?
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  3. #3
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    Oh thanks for your quick reply. I believe it is the group theory proof. Is that the finite abelian group?

    thx
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  4. #4
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    Quote Originally Posted by jenjen View Post
    Oh thanks for your quick reply. I believe it is the group theory proof. Is that the finite abelian group?
    Yes.

    It is an excercise for you (unless you want me to prove it) that the set G_n={1<=x<=n |gcd(x,n)=1} for n>1 is a group under multiplication modulo n. (That is the set of all relative prime elements to n less than n is a group). Since it is finite we have a^m = 1 where m is the number of elements in the group. That is m=phi(n) by definition.
    Q.E.D.

    Now, this group is isomorphic to Z/nZ (you know the classes of integers modulo n) thus, this is true for all integers relatively prime to n.
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