2004 Schweitzer problem 4

Question

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 $f(r)<f(b)$.

2004 Schweitzer Miklós Problem 4

So far, I've learned that

1. $ f(0)=0 $
2. $ f(1)=f(-1)=1 $
3. $f(x) = f(-x)$
4. $ f(n) \ge \log_{2} ( n+1 ) $ for $n > 0$

And of course guess the answer to be $f(n) = |n|^c$ where $c \in \mathbb{Z^*}$

Any other clue to this problem?