This problem was largely inspired by this problem here.
There were many counterexamples given to the problem, such as a multiplicative function that maps primes to a permutation thereof.
However, if $f(x)$ is a strictly increasing function, then what happens then?
It appears that if such $f(x)$ existed, then $f(1)=\gcd(f(1),f(y))$. This implies that $f(1)$ would have to divide $f(x)$ for any integer $x$. So one thing I tried doing was setting $g(x)=\frac{f(x)}{f(1)}$. However, I was not able to proceed further.
So how does one find all $f(n)$ such that $\gcd(f(x),f(y))=f(\gcd(x,y))$, where $f(x)$ is a strictly increasing function?
Note that all the counterexamples given were not increasing. Any help would be appreciated.