Question 160
Let $f: \mathbb{Z} \rightarrow \{1, 2, \dots, 10^{100}\}$ be a function satisfying
$$
\text{gcd}(f(x), f(y)) = \text{gcd}(f(x), x - y)
$$
for all integers $x$ and $y$. Show that there exist positive integers $m$ and $n$ such that $f(x) = \text{gcd}(m + x, n)$ for all integers $x$.
$$
\text{gcd}(f(x), f(y)) = \text{gcd}(f(x), x - y)
$$
for all integers $x$ and $y$. Show that there exist positive integers $m$ and $n$ such that $f(x) = \text{gcd}(m + x, n)$ for all integers $x$.