1 + 1/2 + 1/3 +...+1/823 = r/s*823.
Without calculating the LHS, prove that r= s (mod 823^3).
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?