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I'm currently working in the following Euler's theorem exercise:

Show that if n is even and $n≥2$ then $\phi(n)≤ n/2$

I'm starting from the point that if $n$ is even at least one of its factors is $2$ but still can't find a way to show the required fact, any help will be really appreciated.

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    $\begingroup$ Given that $n$ is even, are there any numbers you can say are definitely not coprime to $n$? $\endgroup$ – Gregory J. Puleo Feb 3 at 3:03
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Since $n$ is even, any even number that's less than or equal $n$ is not coprime to $n$. There are $\frac n2$ of them. The inequality follows.

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Given $n$ is even, all the even numbers less than or equal to $n$ must not be coprime to $n$. There are $\frac n2$ such. $\therefore \phi(n)\le n-\frac n2=\frac n2$.

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For $k\ge1,$

$\phi(2^km)=\phi(2^k)\phi(m)=?$ for odd $m$

Now $\phi(m)\le m-1$

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