Školjka

%V0 Let $a, b$ be integers, and let $P(x) = ax^3+bx.$ For any positive integer $n$ we say that the pair $(a,b)$ is $n$-good if $n | P(m)-P(k)$ implies $n | m - k$ 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. Proposed by Okan Tekman, Turkey