Note on multiplicative functions satisfying a congruence property. II (Q1121309)

[For part I, cf. Acta Sci. Math. 55, 301-308 (1991; Zbl 0666.10001).] Let \(k\) and \(M\) be positive integers and let \(N_ k\) be the arithmetical function defined by \(N_ k(n)=m\), where \(n=m^ k h\), with \(h\) \(k\)-free. Extending a result of \textit{B. M. Phong} and \textit{J. Fehér} [Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 33, 261-265 (1990; Zbl 0749.11007)] the author proves the following result. If \(f\) is an integer- valued multiplicative arithmetical function satisfying \(f(M)\neq 0\) and \(f(n+M)\equiv f(M)\pmod {N_ k(n)}\) for every positive integer \(n\), then there exists a non-negative integer \(a\) such that \(f(n)=n^ a\) for every positive integer \(n\).