# There exist infinitely many positive integers $k>r!$, such that for any $j$ with $r!<j<k$ we have $\prod_{i=0}^{r-1}(j-i)\nmid\prod_{i=0}^{r-1}(k-i)$

Question:

Let $r$ be a postive integer. Show that there exist infinitely many positive integers $k$ satisfying $k>r!$,such that for any positive integer $j$ satisfying $r!<j<k$ we have $$j(j-1)(j-2)\cdots(j-r+1) \nmid k(k-1)(k-2)\cdots(k-r+1).$$

Maybe this is a Mathematical olympiad problem, and I can't prove it. Thank you.

I think this problem can be solved by considering $v_{p}(a)$, the exponent of the prime number $p$ in the prime decomposition of the product $a$ above.