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April 9th, 2018, 07:53 AM  #1 
Newbie Joined: May 2017 From: Moscow Posts: 6 Thanks: 0  Euler's totient function equation
$\displaystyle \varphi(8x)=96$ How do I find x?

April 9th, 2018, 08:57 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 19,865 Thanks: 1833 
Does x have to be a natural number?

April 9th, 2018, 09:10 AM  #3 
Newbie Joined: May 2017 From: Moscow Posts: 6 Thanks: 0 
Yes, sorry forgot to specify that.

April 9th, 2018, 12:35 PM  #4 
Senior Member Joined: Sep 2016 From: USA Posts: 502 Thanks: 280 Math Focus: Dynamical systems, analytic function theory, numerics 
Consider cases for divisibility of $x$ by $2^k$ for $k = 0,1,2...$ and use the fact that $\varphi$ is multiplicative for coprime factors. Example: Suppose $x$ is odd (i.e.\ $k= 0$), then gcd$(8,x) = 1$ so you must have \[\varphi(8x) = \varphi( 8 ) \varphi(x) = 96 \implies \varphi(x) = 24 \] Solve this by considering the decomposition of $x$ into prime powers and apply the multiplicative property once again. Repeat for the case that $x = 2^kn$ for $n$ odd with increasing $k$ and it won't take long to conclude all possible values of $k$. Do you see why? 

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equation, euler, function, totient 
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