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Math Help - euler phi problem

  1. #1
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    euler phi problem

    let p be prime, show that p does not divide n where n is a positive integer if and only if phi(np) = (p-1)phi(n)

    thanks in advance.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Whenever (a,b)=1 we have \phi(ab)=\phi(a)\phi(b). So if p \nmid n, then (p,n)=1 and \phi(pn)=\phi(p)\phi(n)=(p-1)\phi(n).

    Conversely suppose \phi(pn)=(p-1)\phi(n); write n=p^\alpha m,\: p \nmid m. Then \phi(n)=\phi(p^\alpha)\phi(m) and \phi(pn)=\phi(p^{\alpha+1})\phi(m)=\frac{\phi(p^{\  alpha+1})}{\phi(p^\alpha)}\phi(n). From this and our hypothesis we get \frac{\phi(p^{\alpha+1})}{\phi(p^\alpha)} = p-1. But if \alpha>0, \frac{\phi(p^{\alpha+1})}{\phi(p^\alpha)} = \frac{p^{\alpha+1}(1-1/p)}{p^{\alpha}(1-1/p)} = p, which contradicts \frac{\phi(p^{\alpha+1})}{\phi(p^\alpha)} = p-1; hence \alpha=0 and p \nmid n.
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