Let $A_{n}$ be alternating group of degree $n$. It's known if $p$ is the greatest prime not exceeding $n$, then the number of Sylow $p$-subgroup of $ A_{n}$ is $n!/p(p-1)(n-p)!$.
I would like to know if $r$ is a arbitary prime divisor of order group $A_{n}$, then what is the number of Sylow $r$ -subgroup of $A_{n}$?
Also I would like to know whether the number of Sylow $r$-subgroup is a multiple of $p$?
All thoughts appreciated!