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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