Question 132
Find all functions $f: \mathbb{Z}^+ \to \mathbb{Z}^+$ such that the following conditions hold:
(i) $a - b \mid f(a) - f(b)$ for all positive integers $a \neq b$,
(ii) $f(\varphi(a)) = \varphi(f(a))$ for all $a \in \mathbb{Z}^+$.
Note. $\varphi$ is Euler's totient function: $\varphi(n)$ is the number of positive integers not exceeding $n$ and coprime with $n$.
(i) $a - b \mid f(a) - f(b)$ for all positive integers $a \neq b$,
(ii) $f(\varphi(a)) = \varphi(f(a))$ for all $a \in \mathbb{Z}^+$.
Note. $\varphi$ is Euler's totient function: $\varphi(n)$ is the number of positive integers not exceeding $n$ and coprime with $n$.