Question: > 
> Let $N>1$ be a positive integer. What will be the Arithmetic mean of positive integers less than $N$ and co-prime with $N$?
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Getting no idea how to proceed!

Answer: 
(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 $10$.
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](http://en.wikipedia.org/wiki/Euler%27s_totient_function) function, the number of positive integers $<n$ and co-prime to $n$

The Arithmetic mean will be $\frac S{\phi(n)}$