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Math Help - Multiplicative inverse properties

  1. #1
    Super Member Bacterius's Avatar
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    Multiplicative inverse properties

    Hello,
    I just have a question regarding multiplicative inverses.

    Let a \equiv p_1 \times p_2 \times \cdots \times p_n \pmod{x} where p_i is a prime and x is any positive integer and \gcd(x, a) = 1. Does the following hold :

    a^{-1} \equiv p_1^{-1} \times p_2^{-1} \times \cdots \times p_n^{-1} \pmod{x}

    Thank you
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Yes! The group (\mathbb{Z}/x\mathbb{Z})^\times of invertible residue classes \mod x is an abelian group, and a=a_1a_2 \Rightarrow a= a_1^{-1}a_2^{-1}. The same holds for a=a_1\dots a_n.
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  3. #3
    Super Member Bacterius's Avatar
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    Quote Originally Posted by Bruno J. View Post
    Yes! The group (\mathbb{Z}/x\mathbb{Z})^\times of invertible residue classes \mod x is an abelian group, and a=a_1a_2 \Rightarrow a= a_1^{-1}a_2^{-1}. The same holds for a=a_1\dots a_n.
    Thanks Bruno!
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