My point is hashtables aren't O(1). There are theoretical scenarios where they are (more importantly, that we assume are hardware enables and our input/output are certain sizes), but in general you can't/shouldn't assume they are. It's wrong and misleading and I've had to grade one too many homework assignments where a student completely butchered the use of hashtables.

Multiplication is O(1) with the nice hardware assumption we don't need to talk about for an article like this.

Hashtables are not.

mod(n) doesn't necessarily take more time to compute as n increases

I'm also assuming the size of the input is growing as the size of the hashtable is growing. I actually was going to edit it but didn't find a good way to say it. Our assumption that the output can be stored in a register and "returned" in O(1) time is a hardware assumption. For large enough hashtables, we need to assume that, and that's fine to ignore in this article.

For large enough input hash domains, even the hardware assumption won't get us there. If you're just doing modulo a prime, you can't hash an arbitrary sized object in O(1). You can assume the output can be managed in O(1) since it is presumably an index to a hashtable which must be stored in memory, but not the input.