CMO 2012 Problem 2

For any positive integers $n$ and $k$, let $L(n, k)$ be the least common multiple of the $k$ consecutive integers $n, n+1, \ldots, n+k-1$. Show that for any integer $b$, there exist integers $n$ and $k$ such that $L(n, k)>b L(n+1, k)$.