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

Question

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.

Answer 1

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.

Answer 2

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. $\therefore \phi(n)\le n-\frac n2=\frac n2$.

Answer 3

For $k\ge1,$

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

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