Problem

Source: iran2004(number theory exam)

Tags: number theory proposed, number theory



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)$.