RU-ALLRUSSIAN 2009 Problem 93

Given are positive integers $n>1$ and $a$ so that $a>n^2$ , and among the integers $a+1, a+2, \ldots, a+n$ one can find a multiple of each of the numbers $n^2+1, n^2+2, \ldots, n^2+n$ . Prove that $a>n^4-n^3$ .