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Thread: Phi Function

  1. #1
    Super Member
    Feb 2008

    Phi Function

    Let & denote the phi function

    Show that if m and n are not relatively prime, then &(m)&(n) < &(mn)
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  2. #2
    Super Member PaulRS's Avatar
    Oct 2007
    Well, if (m,n)>1 then they must have a common prime divisor p.

    Let p^{\alpha} and p^{\beta} be the greatest powers of p dividng m and n respectively, then p^{\alpha+\beta} is the greatest power of p dividng m\cdot n.

    But then \phi\left(p^{\alpha}\right)\cdot \phi\left(p^{\beta}\right) < \phi\left(p^{\alpha+\beta}\right) and you can finish it off by remembering \phi is multiplicative.
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