Weird GCD Value

You are given two positive integers $a$ and $b$, we define $f(a, b)$ as the maximum value of $|\gcd(a, x) - \gcd(b, x)|$ where $x$ is some natural number. Formally, $f(a, b) = \max\limits_{x \in \mathbb{N}}| \gcd(a,x) - \gcd(b,x)|$ , where $\mathbb{N}$ represents the set of natural numbers and $\gcd(a, b)$ represents the greatest common divisor of $a$ and $b$. You are given an integer $k$. Y