Question 57
Let $F$ be the set of all functions $f: \mathbb{Z} \setminus \{0\} \to \mathbb{N}^*$ with the following property: If $a, b \in \mathbb{Z} \setminus \{0\}$ and $a$ is not divisible by $b$ then there exist integers $r, s$ such that $a = br + s$ and $f(s) \lt f(b)$.
Find all functions $f_0 \in F$ such that $\forall f \in F, \forall n \in \mathbb{Z} \setminus \{0\}$, we have $f_0(n) \le f(n)$.
Find all functions $f_0 \in F$ such that $\forall f \in F, \forall n \in \mathbb{Z} \setminus \{0\}$, we have $f_0(n) \le f(n)$.