The function $\psi : \mathbb{N}\rightarrow\mathbb{N}$ is defined by $\psi (n)=\sum_{k=1}^n\gcd (k,n)$. $(a)$ Prove that $\psi (mn)=\psi (m)\psi (n)$ for every two coprime $m,n \in \mathbb{N}$. $(b)$ Prove that for each $a\in\mathbb{N}$ the equation $\psi (x)=ax$ has a solution.