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Find all positive integers $k$

Source: 6-th Taiwanese Mathematical Olympiad 1997

January 17, 2007
functionnumber theory proposednumber theory

Problem Statement

Find all positive integers kk for which there exists a function f:N→Zf: \mathbb{N}\to\mathbb{Z} satisfying f(1997)=1998f(1997)=1998 and f(ab)=f(a)+f(b)+kf(gcd⁡(a,b))∀a,bf(ab)=f(a)+f(b)+kf(\gcd{(a,b)})\forall a,b.
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