IMO 2010 — Complete Shortlist Question 26

Let $a, b$ be integers, and let $P(x)=a x^{3}+b x$. For any positive integer $n$ we say that the pair $(a, b)$ is $n$-good if $n \mid P(m)-P(k)$ implies $n \mid m-k$ for all integers $m, k$. We say that $(a, b)$ is very good if $(a, b)$ is $n$-good for infinitely many positive integers $n$.
(a) Find a pair $(a, b)$ which is 51 -good, but not very good.
(b) Show that all 2010-good pairs are very good.