Find all functions satisfying $\frac{1}{f(a)}+ \frac{1}{f(b)} = \frac{1}{f(c)} $ whenever $\frac{1}{a}+ \frac{1}{b} = \frac{1}{c} $

Hint: Try to show that $f(ab) = af(b)$ for all $a,b\in\mathbb{N}$. This can be done through induction on $a$: Suppose $f(kb) = kf(b)$ for all $b\in\mathbb{N}$ and $k = 1,\dots,a-1$, and note that $$\frac{1}{ab} + \frac{1}{(a-1)\times ab} = \frac{1}{(a-1)\times b}. $$ Can you proceed from here?