multiplicative function - gcd - lcm

Question

How can I show if $f$ is multiplicative and $x,y \in \mathbb{Z}_{>0}$, that this implies that $f (\text{gcd} (x,y)) * f ( \text{lcm} (x,y)) = f(x)*f(y)$?

Thank you very much.

Answer 1

Since $f$ is multiplicative, it is only necessary to show this for prime powers. So let $p$ be a prime and let $i,j$ be non-negative integers and take $x=p^i$ and $y=p^j$. We know that $\gcd(x,y)=\min(x,y)$ and $\textrm{lcm}(x,y)=\max(x,y)$, so $f (\text{gcd}(x,y)) \cdot f (\text{lcm} (x,y)) = f(x)\cdot f(y)$ follows immediately. Therefore, the result holds for all positive integer values of $x$ and $y$.