Prove that if $n>1$, the sum of positive integers less than $n$ and coprime to $n$ is $(1/2)na(n)$ where $a(n)$ is the number of such integers. [duplicate]

Question

Question 12(iii) Could anyone explain this part of the question to me.

What i tried co-prime means that the two integers a and b are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1

Take th number $3$ for example, then the sum of integers less than $3$ and co-prime to $3$ is $2+1=3$, $2$ and $1$ are the two integers co-prime to $3$ which thus satisfies the formula $0.5*n*a(n)$ where $n=3$, $a(n)=2$. However im unsure of how to prove it. Could anyone explain. Thanks

Answer 1

Hint

You have already shown that $$ (m,n)=1 \iff (m,m-n)=1$$

See if you can figure out what is going on with the sum of such integers.

By the way, the notation for the Euler function is $\phi (n)$ not $\alpha (n)$

Answer 2

Hint:

If $m$ is coprime to $n$ then $n-m$ is also coprime to $n$.