Question 25
For positive integers $n$ and $k \geqslant 2$ define $E_{k}(n)$ as the greatest exponent $r$ such that $k^{r}$ divides $n$ !. Prove that there are infinitely many $n$ such that $E_{10}(n)\gt E_{9}(n)$ and infinitely many $m$ such that $E_{10}(m)\lt E_{9}(m)$.