BG-MO 2023 Problem 3

Let $f(x)$ be a polynomial with positive integer coefficients. For every $n\in\mathbb{N}$ , let $a_{1}^{(n)}, a_{2}^{(n)}, \dots , a_{n}^{(n)}$ be fixed positive integers that give pairwise different residues modulo $n$ and let $$g(n) = \sum\limits_{i=1}^{n} f(a_{i}^{(n)}) = f(a_{1}^{(n)}) + f(a_{2}^{(n)}) + \dots + f(a_{n}^{(n)})$$ Prove that there exists a constant $M$ such that for all integers $m>M$ we have $\gcd(m, g(m))>2023^{2023}$ .