Question 25
The function $ \psi $ from the set N of positive integers into itself is defined by the equality
$$ \psi(n)=\sum_{k=1}^{n}(k,n),\qquad n\in\mathbb{N}, $$
where $ (k,n) $ denotes the greatest common divisor of k and n.
(a) Prove that $ \psi(mn) = \psi(m)\psi(n) $ for every two relatively prime $ m, n \in \mathbb{N} $.
(b) Prove that for each $ a \in \mathbb{N} $ the equation $ \psi(x) = ax $ has a solution.
(c) Find all $ a \in \mathbb{N} $ such that the equation $ \psi(x) = ax $ has a unique solution.
$$ \psi(n)=\sum_{k=1}^{n}(k,n),\qquad n\in\mathbb{N}, $$
where $ (k,n) $ denotes the greatest common divisor of k and n.
(a) Prove that $ \psi(mn) = \psi(m)\psi(n) $ for every two relatively prime $ m, n \in \mathbb{N} $.
(b) Prove that for each $ a \in \mathbb{N} $ the equation $ \psi(x) = ax $ has a solution.
(c) Find all $ a \in \mathbb{N} $ such that the equation $ \psi(x) = ax $ has a unique solution.