SCHWEITZER 2004 Problem 4

Determine all totally multiplicative and non-negative functions $f\colon\mathbb{Z}\rightarrow \mathbb{Z}$ with the property that if $a, b\in \mathbb{Z}$ and $b\neq 0$ , then there exist integers $q$ and $r$ such that $a-qb+r$ and $f(r)<f(b)$ .