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# 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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