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Math Help - Order of integers and primitive roots

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    Order of integers and primitive roots

    Let a,m be in Z with m>0. If a' is the inverse of a modulo m, prove that the order of a modulo m is equal to the order of a' modulo m. Deduce that if r is a primitive root modulo m, then r' is a primitive root modulo m.
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    Senior Member Tinyboss's Avatar
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    For the first part, note that (a^{-1})^n=(a^n)^{-1} for all n.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by meshel88 View Post
    Let a,m be in Z with m>0. If a' is the inverse of a modulo m, prove that the order of a modulo m is equal to the order of a' modulo m. Deduce that if r is a primitive root modulo m, then r' is a primitive root modulo m.
    The first part as Tinyboss noted follows easily from consider elements of \left(\mathbb{Z}/n\mathbb{Z}\right)^{\times}
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