2. Polynomials $P(x)$ and $Q(x)$ of equal degree are called similar if one can be obtained from the other by permuting the coefficients (for example, the polynomials $2 x^{3}+x+7$ and $x^{3}+2 x^{2}+7 x$ are similar). For what largest $k$ is it true that for any similar polynomials $P(x), Q(x)$, the number $P(2009)-Q(2009)$ is necessarily divisible by $k$?
2. Answer: 2008.

Note that for similar polynomials, the sums of the coefficients coincide, that is, $P(1)=Q(1)$. Then 1 is a root of the polynomial $P(x)-Q(x)$, which means $P(x)-Q(x)=(x-1) \cdot R(x)$, where $R(x)$ is also a polynomial with integer coefficients. Then $P(2009) - Q(2009) = 2008 \cdot R(2009)$. On the other hand, for $P(x)=x$ and $Q(x)=1$, it is clear that a larger $k$ does not exist.