Question 13
Let $\mathbb{Z}$ denote the set of integers and let $S \subset \mathbb{Z}$ be the set of integers that are at least $10^{100}$. Fix a positive integer $c$. Determine all functions $f: S \to \mathbb{Z}$ satisfying
$$
f(xy + c) = f(x) + f(y) \quad \text{for all } x, y \in S.
$$
$$
f(xy + c) = f(x) + f(y) \quad \text{for all } x, y \in S.
$$