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Math Help - Euler phi function

  1. #1
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    Euler phi function

    hi,

    I need help
    let gcd(n, m) = 1 prove that \phi(nm)=\phi(n)\phi(m)

    thanks
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  2. #2
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    Quote Originally Posted by asub1 View Post
    hi,

    I need help
    let gcd(n, m) = 1 prove that \phi(nm)=\phi(n)\phi(m)

    thanks
    Define a function, \phi : \mathbb{Z}_{nm}^{\times} \to \mathbb{Z}_n^{\times}\times \mathbb{Z}_m^{\times} by \phi ([x]_{nm}) = ([x]_n,[x]_m).
    Now argue that this mapping is one-to-one and onto by the Chinese remainder theorem.
    And so it follows that |\mathbb{Z}_{nm}^{\times}| = |\mathbb{Z}_n^{\times}| \times |\mathbb{Z}_m^{\times} | \implies \phi(nm) = \phi(n)\phi(m).
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  3. #3
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    Thank you very much!
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  4. #4
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    Thank you very much again
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