IR-TST 2021 Problem 3

There exist $4$ positive integers $a,b,c,d$ such that $abcd \neq 1$ and each pair of them have a GCD of $1$ . Two functions $f,g : \mathbb{N} \rightarrow \{0,1\}$ are multiplicative functions such that for each positive integer $n$ we have : $$f(an+b)=g(cn+d)$$ Prove that at least one of the followings hold. $i)$ for each positive integer $n$ we have $f(an+b)=g(cn+d)=0$ $ii)$ There exists a positive integer $k$ such that for all $n$ where $(n,k)=1$ we have $g(n)=f(n)=1$ (Function $f$ is multiplicative if for any natural numbers $a,b$ we have $f(ab)=f(a)f(b)$ )

Proposed by Navid Safaii