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Part of 2012 Federal Competition For Advanced Students, Part 1

Problems(1)

Function such that (m,n)|f(m)+f(n)

Source: Austrian Federal Competition for Advanced Students 2012, #1

5/23/2012
Determine all functions f:Z→Zf: \mathbb{Z}\to\mathbb{Z} satisfying the following property: For each pair of integers mm and nn (not necessarily distinct), gcd(m,n)\mathrm{gcd}(m, n) divides f(m)+f(n)f(m) + f(n).
Note: If n∈Zn\in\mathbb{Z}, gcd(m,n)=gcd(∣m∣,∣n∣)\mathrm{gcd}(m, n)=\mathrm{gcd}(|m|, |n|) and gcd(n,0)=n\mathrm{gcd}(n, 0)=n.
functionnumber theorygreatest common divisorgeometrygeometric transformationnumber theory proposed