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Euclidian algorithm-ish function

Source: 2022 IRN TWN Friendly math competition P6

June 23, 2022
functionnumber theoryalgebraIranTaiwan

Problem Statement

Find all completely multipiclative functions f:Z→Z⩾0f:\mathbb{Z}\rightarrow \mathbb{Z}_{\geqslant 0} such that for any a,b∈Za,b\in \mathbb{Z} and b≠0b\neq 0, there exist integers q,rq,r such that a=bq+ra=bq+r and f(r)<f(b)f(r)<f(b)
Proposed by Navid Safaei
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