Question 14
The real-valued function $f$ is defined for all positive integers. For any integers $a \gt 1$, $b \gt 1$ with $d = \operatorname{gcd}(a, b)$, we have
$$
f(ab) = f(d) \cdot \left(f\left(\frac{a}{d}\right) + f\left(\frac{b}{d}\right)\right),
$$
Determine all possible values of $f(2001)$.
$$
f(ab) = f(d) \cdot \left(f\left(\frac{a}{d}\right) + f\left(\frac{b}{d}\right)\right),
$$
Determine all possible values of $f(2001)$.