Put another way,
How do I show that p^3 divides r-s in this instance?
Well, if , then . Since is prime, by Wolstenholme's Theorem, the numerator of is divisible by , and by definition, is the lowest positive integer ensuring the RHS is a whole number, essentially doing the job of canceling all the denominators: , for some and .
Therefore, the entire RHS is divisible by , hence
*Are you asking us to prove Wolstenholme's Theorem or to simply connect the dots enough to make use of this theorem?