Replacement rule to set `Floor[Mod[H, p1*p2...pm]/(p1*p2...pm)]` to zero where the length of the product p1*p2...pm will vary

Question

For example I have tried:

Floor[Mod[H, p1*p2*p3]/(p1*p2*p3)]/.{Floor[Mod[H, x_]/(x_)]->0}

and

Floor[Mod[H, p1*p2*p3]/(p1*p2*p3)]/.{Floor[Mod[H, x__]/(x__)]->0}

but they not does work except for the case where the product is just p1. The number of elements in these products will vary from p1 to p1*p2*...*pm.

Answer 1

The thing to keep in mind is that / is translated into Times and Power, which one can see in the FullForm:

Floor[Mod[H, p1*p2*p3]/(p1*p2*p3)] // FullForm
(*
  Floor[Times[Power[p1, -1], Power[p2, -1], Power[p3, -1],  Mod[H, Times[p1, p2, p3]]]]
*)

So to match the denominator with the argument of Mod takes something like this:

Floor[Mod[H, p1*p2*p3]/(p1*p2*p3)] /. {Floor[
    Mod[H, x_] * y__ /; 1/Times[y] === x] -> 0}
(*  0  *)