ÖMO 2008 Problem 2

(a) Does there exist a polynomial $P(x)$ with coefficients in integers, such that $P(d) = \frac{2008}{d}$ holds for all positive divisors of $2008$ ? (b) For which positive integers $n$ does a polynomial $P(x)$ with coefficients in integers exists, such that $P(d) = \frac{n}{d}$ holds for all positive divisors of $n$ ?