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.

Given that $n$ is even, are there any numbers you can say are definitely not coprime to $n$? — Gregory J. Puleo, Feb 03 '19 at 03:03

score 4 · Accepted Answer · answered Feb 03 '19 at 03:09

4

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.

answered Feb 03 '19 at 03:09

abc...

5,010

score 2 · Answer 2 · answered Feb 03 '19 at 03:13

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

answered Feb 03 '19 at 03:13

score 2 · Answer 3 · answered Feb 03 '19 at 03:23

2

For $k\ge1,$

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

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

answered Feb 03 '19 at 03:23

lab bhattacharjee

279,016

Do you mean in the second line "for even $m$"? – mraz Feb 03 '19 at 03:28
1

@mraz, No, I've used https://math.stackexchange.com/questions/192452/whats-the-proof-that-the-euler-totient-function-is-multiplicative – lab bhattacharjee Feb 03 '19 at 03:40
Got it, thank you – mraz Feb 03 '19 at 03:55

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

3 Answers3