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

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

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