TITLE: Divisors of factorials
QUESTION [5 upvotes]: Let $p$ be a prime number and $k$ a positive integer. Let $d$ be the smallest positive integer such that $p^k$ divides $d!$. It is true that $d$ is necessarily a multiple of $p$?

REPLY [7 votes]: Suppose $d!$ is this number. Then $(d-1)!$ is not. Since $p^k$ does not divide $(d-1)!$ but it divides $d! = d \cdot (d-1)!$, then $p$ must divide $d$. 
Hope that helps,