Question 106
Let $S$ denote the set of polynomials $P(x) = a x^2 + b x$ where $0 \leq a, b \lt 2010^{18}$ are integers. Determine the number of polynomials $P$ in $S$ for which there exists a polynomial $Q$ in $S$ such that $Q(P(n)) \equiv n \pmod{2010^{18}}$ for all integers $n$.