Euler phie function

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  • #60440

    If 0<n<1000. Find all n such that it minimizes phie(n)/n.( n is a natural number)

     

    #60489
    Saumik Karfa
    Moderator

    We will get back to you within 24 hours.

    #60726
    Saumik Karfa
    Moderator

    Case 1 :  Let $n=1$

    $\frac {\phi (n)}{n}=1$

    Case 2 : Let $n \ne 1$

    then $n= p_1^{a_1}p_2^{a_2}\ldots p_k^{a_k}$ where $p_i’$s are distinct primes and  $a_i\in \mathbb N,\quad i=1,2,\ldots k$

    then $\frac{\phi(n)}{n}=\frac{(p_1-1)(p_2-1)\ldots(p_k-1)}{p_1 p_2\ldots p_k} < 1$

    hence cannot be an integer.

    ANS : Only $n=1$

     

    #60743

    <p style=”text-align: center;”>It is stated that it minimizes not maximizes. if phi(n)/n  has to maximize than it must be 1. But function phi(n)/n has to minimize. So is not the answer . I think the answers are 210,420,630.</p>

    #60805
    Saumik Karfa
    Moderator

    Sorry I misread the problem. Thank you pointing it out. Will get back to you soon

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