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Math Help - observations of Gn (invertible classes of Zn)

  1. #1
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    observations of Gn (invertible classes of Zn)

    Choose a value of n and count the number of elements in Gn (Gn is the set of invertible congurence classes of Zn). Can you discover any rules between n and the number of elements in Gn?

    I have so far that if n is prime then [1]n, [2]n, ... , [n-1]n are all in Gn

    also if n = 2^m for some integer m, Gn contains all odd elements of Zn

    and finally, I know the product of any two elements of Gn is in Gn, for
    n >= 2

    what conclusions can I draw knowing only this information? What have I missed?
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    Quote Originally Posted by minivan15 View Post
    Choose a value of n and count the number of elements in Gn (Gn is the set of invertible congurence classes of Zn). Can you discover any rules between n and the number of elements in Gn?

    I have so far that if n is prime then [1]n, [2]n, ... , [n-1]n are all in Gn

    also if n = 2^m for some integer m, Gn contains all odd elements of Zn

    and finally, I know the product of any two elements of Gn is in Gn, for
    n >= 2

    what conclusions can I draw knowing only this information? What have I missed?
    Let 0 < a < n. Then a is invertible iff ax\equiv 1(\bmod n) for some x. But this congruence is solvable if and only if (a,n)=1. Thus, all elements relatively prime to n are invertible, there are \phi(n) such elements.
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