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April 9th, 2018, 06:53 AM   #1
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Euler's totient function equation

$\displaystyle \varphi(8x)=96$ How do I find x?
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April 9th, 2018, 07:57 AM   #2
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Does x have to be a natural number?
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April 9th, 2018, 08:10 AM   #3
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Yes, sorry forgot to specify that.
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April 9th, 2018, 11:35 AM   #4
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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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