Question 77
For a prime number $p$ and a positive integer $n$, denote by $f(p, n)$ the largest integer $k$ such that $p^k \mid n!$. Let $p$ be a given prime number and let $m$ and $c$ be given positive integers. Prove that there exists infinitely many positive integers $n$ such that $f(p, n) - c$ is divisible by $m$.