Let $f(n)$ be the function that sums all positive integers that are coprime to $n$. For example, if we take $n=6$, $f(6)=1+5=6$ because $\gcd(1,6)=\gcd(5,6)=1$.
The formula for $f(n)$ is given as $f(n)=n\frac{\phi(n)}{2}$, where $\phi(n)$ denotes the Euler phi function. By this formula and the multiplicativity of $\phi(n)$, we say $f(nm)=2f(n)f(m)$.
My question is, is there a way to obtain $f(nm)=2f(n)f(m)$ without applying $f(n)$'s formula?