Nicolae's function

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.

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