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Euler's Totient Function Proving

  1. Feb 1, 2010 #1
    I need some help/hints on how to prove this statement. I don't know where to start!


    Prove that if m and n are positive integers such that m|n, then φ(mn) = mφ(n).

    Thanks
     
  2. jcsd
  3. Feb 1, 2010 #2
    It's a direct consequence of Euler's formula:
    [tex]
    \phi\left(n\right)=n(1-\frac{1}{p_1})(1-\frac{1}{p_2}) \cdots (1-\frac{1}{p_k})
    [/tex]
    Because the hypothesis implies that the prime factorization of m has the same factors as n.

    Or you can count the numbers that are coprime with nm in [tex]\left\{1,\ldots,mn\right\}[/tex] and prove that there are as many as the ones that are coprime with n in [tex]\left\{1,\ldots,n\right\}[/tex].
     
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