Does This Polynomial Even Exist?

Number Theory Level 5

Find the number of positive integers \(n\) such that \(2 \leq n \leq 100\) and there exists a polynomial \(f(x)\) with real coefficients and degree \( <n\) such that for all integers \(x,\) \(f(x)\) is an integer if and only if \(x\) is not a multiple of \(n.\)

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