OFM 2021 S4

Let $\mathbb{N}_{\ge 1}$ be the set of positive integers. Find all functions $f \colon \mathbb{N}_{\ge 1} \to \mathbb{N}_{\ge 1}$ such that, for all positive integers $m$ and $n$ :

(a) $n = \left(f(2n)-f(n)\right)\left(2 f(n) - f(2n)\right)$ , (b) $f(m)f(n) - f(mn) = \left(f(2m)-f(m)\right)\left(2 f(n) - f(2n)\right) + \left(f(2n)-f(n)\right)\left(2 f(m) - f(2m)\right)$ , (c) $m-n$ divides $f(2m)-f(2n)$ if $m$ and $n$ are distinct odd prime numbers.