Nicolae's function

Number Theory Level 4

For how many integer values $k$ does there exist a function $f$ from the positive integers to the integers such that $f(135)=3$ and for all pairs of positive integers $(m,n)$ ,

$f(mn)=f(m)+f(n)+k\times f(gcd(m,n)).$

This problem is shared by Nicolae S. from the Czech and Slovak Republic Olympiad.