Arithmetic mean of positive integers less than an integer $N$ and co-prime with $N$. 
Let $N>1$ be a positive integer. What will be the Arithmetic mean of positive integers less than $N$ and co-prime with $N$?

Getting no idea how to proceed!
 A: (The way the problem reads at this time is "What will be the Arithmetic mean of positive integers less than $N$ and co-prime with $N$?" (I mention this in case there was some other version of the question and it got edited.))
$n$ is coprime to $N$ if and only if $N-n$ is coprime to $N$.
The average of $n$ and $N-n$ is $N/2$.
The average of the averages of all such cases is the average of a bunch of numbers each equal to $N/2$.
The average of all numbers in $\{1,\dots,N\}$ that are coprime to $N$ is therefore $N/2$.
Later edit: Here's an example.  The numbers in $\{1,\dots,20\}$ that are coprime to $20$ are $1,3,7,9,11,13,17,19$.
They come in these pairs:
\begin{align}
1, & 19 \text{ (The average of these two is $10$.)} \\
3, & 17 \text{ (The average of these two is $10$.)} \\
7, & 13 \text{ (The average of these two is $10$.)} \\
9, & 11 \text{ (The average of these two is $10$.)}
\end{align}
Now take the average of all those $10$s.  The average is $10$.
A: HINT:
As $(a,n)=(n-a,n),$
Let $$S=\sum_{1\le a\le n,(a,n)=1}a,$$
then $$S=\sum_{1\le a\le n,(a,n)=1}(n-a)$$
$$\implies2S=\phi(n)(a+n-a)=n\phi(n)$$ where $\phi(n)$ is the Euler Totient function, the number of positive integers $<n$ and co-prime to $n$
The Arithmetic mean will be $\frac S{\phi(n)}$
