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Let $f(x)$ be a polynomial with integer coefficients. Let us assume that there exists a positive integer $k$ and $k$ consecutive integers $n, n+1, \ldots, n+k-1$ such that none of the numbers $f(n), f(n+1), \ldots, f(n+k-1)$ is divisible by $k$. Prove that the roots of $f(x)$ are not integers.
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