Question 16
We denote the set of all nonzero integers and the set of all nonnegative integers by $\mathbb{Z}^*$ and $\mathbb{N}_0$, respectively. Find all functions $f:\mathbb{Z}^* \to \mathbb{N}_0$ for which the following two conditions hold:
(1) for each $a,b \in \mathbb{Z}^*$ such that $a+b \in \mathbb{Z}^*$ it holds that $f(a+b) \ge \min\{f(a),f(b)\}$;
(2) for each $a,b \in \mathbb{Z}^*$ it holds that $f(ab)=f(a)+f(b)$.
(1) for each $a,b \in \mathbb{Z}^*$ such that $a+b \in \mathbb{Z}^*$ it holds that $f(a+b) \ge \min\{f(a),f(b)\}$;
(2) for each $a,b \in \mathbb{Z}^*$ it holds that $f(ab)=f(a)+f(b)$.