TSTST 2019 Problem 7

Let $f: \mathbb{Z} \rightarrow\left\{1,2, \ldots, 10^{100}\right\}$ be a function satisfying $$\operatorname{gcd}(f(x), f(y))=\operatorname{gcd}(f(x), x-y)$$ for all integers $x$ and $y$. Show that there exist positive integers $m$ and $n$ such that $f(x)=\operatorname{gcd}(m+x, n)$ for all integers $x$.