> (x * p) is repeated addition with p integer Yes, that's the "first bit" that I was not raising an issue with. > dividing by q involves finding r such that r

> (x * p) is repeated addition with p integer Yes, that's the "first bit" that I was not raising an issue with. > dividing by q involves finding r such that r * q = x * p Which, as you note, requires a trial and error algorithm to find r. But in any case, r is the answer being sought, not the thing we're supposed to be adding to itself some number of times to get the answer.