That would appear to be evidence that it is not harder than division? From your edit I guess that you already realized this, but the hard part here is that for

That would appear to be evidence that it is not harder than division? From your edit I guess that you already realized this, but the hard part here is that for 1 / q you would have to perform up to q divisions of integers of size n = log₂(q) which of course takes time O(2ⁿ) even assuming the divisions are O(1).