IR-MO 2004 Problem 14

We define $f: \mathbb{N} \rightarrow \mathbb{N}$ , $f(n) = \sum_{k = 1}^{n}(k,n)$ .

a) Show that if $\gcd(m,n)=1$ then we have $f(mn)=f(m)\cdot f(n)$ ;

b) Show that $\sum_{d|n}f(d) = nd(n)$ .