Problem

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Tags: algebra, polynomial, modular arithmetic, function, number theory, greatest common divisor, system of equations



Determine all positive integers $n$ for which there exists a polynomial $f(x)$ with real coefficients, with the following properties: (1) for each integer $k$, the number $f(k)$ is an integer if and only if $k$ is not divisible by $n$; (2) the degree of $f$ is less than $n$. (Hungary) Géza Kós