Question 82
For positive integers $n$ and $k \ge 2$ 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 $E_{10}(m) \lt E_9(m)$.