Question

Let n be a fixed positive integer and S be the set of all positive
integers which are less than ‘ n ’ and relatively prime to it. What is the sum
of all the elements of S .

Answer 1

Let $a_1,a_2,\ldots,a_{\phi(n)}$ be the positive integers less than $n$ and relatively prime to $n$.

Since $\gcd(a,n)=\gcd(n-a,n)$, the numbers  $a_1,a_2,\ldots,a_{\phi(n)}$ are same as the numbers  $n-a_1,n-a_2,\ldots,n-a_{\phi(n)}$ in some order. Thus,
$\sum\limits_{i=1}^{\phi(n)}a_i=\sum\limits_{i=1}^{\phi(n)}(n-a_i)=\phi(n)n-\sum\limits_{i=1}^{\phi(n)}a_i$

$\implies \sum\limits_{i=1}^{\phi(n)}a_i=\frac{n\phi(n)}{2}$