What do weighing scales and algebraic number theory have in common?

How would you compare the weight of 5 objects at the same time? More precisely, how can you decide whether the objects all have equal weight or not in the most efficient way? There is a device you can build, similar to the traditional scale, that should be just the right tool... at least intuitively. But if you want to prove that it solves our problem, then some unexpected tools come in very handy: complex numbers and polynomials! It turns out that those exercises in polynomial division you might have done in school might be useful in the real world after all. Caveat: As pointed out in the comments, the conclusion only holds for commensurable weights (i.e. rational in a certain unit of measurement). This would happen in an ideal world where our rational approximations to weights are the actual weights of objects. But... the real world might be slightly more complicated and I’m not a physicist. I just wanted to show you some nice maths :). Maybe you have a suggestion for presenting the problem differently.