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

Thank you soo much in advance!

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- Feb 24th 2007, 03:08 PMjenjenEuler's theorem
Will anyone please show me a proof of Euler's theorem.

Thank you soo much in advance! - Feb 24th 2007, 03:12 PMThePerfectHacker
- Feb 25th 2007, 12:31 PMjenjen
Oh thanks for your quick reply. I believe it is the group theory proof. Is that the finite abelian group?

thx - Feb 25th 2007, 12:53 PMThePerfectHacker
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.