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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: 20,301 Thanks: 1971 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: 559 Thanks: 324 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? Thanks from topsquark, Adam Ledger and cjem

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