Perhaps I don't understand your question properly, but this looks like en.wikipedia.org/wiki/Chinese_remainder_theorem

@Sune: Fantastic! Now I can't even trust my own sense of `nice'.

@Vectornaut: Great, now I am totally confused :) I am going to undelete my answer. Gowers had convinced me that it was wrong, but now I really am not sure. If people don't like the answer they should mark it down so that it can at least inform people of what not to do!

Is the sunflower spiral the densest packing in $\mathbb{R}^2$ constructed in this manner? That is, by continued scaling and rotation

I think I agree. I can't actually construct a situation where it doesn't seem somehow nicer to add "Now" or "Then" or "Consider when" or something similar.

Well, I guess you get the brownie points then :)

Thanks Robin! I realised this as I was reading about the Smith normal form. There exist polynomial time algorithms to compute the Smith and the Hermite normal forms so this certainly answers my question.