So this is basically the generalization of the Chinese Remainder Theorem to the case of non linear congruences. It didn't seem too hard to understand, but I still have some problems with the Extended Euclidean algorithm, which would be needed to find inverses.
This is a way to attack RSA, and also a caveat to those building an RSA to choose p and q well in the system.
Thinking about this section has made me think that slide rules must be build on modular arithmetic, that the two parts of the slide rule move against each other the same way that the two modular expressions move against each other when you are using the Chinese remainder theorem. Also, you can use a slide rule to take square roots, and you can use the Chinese remainder theorem to take square roots.
There are geometric interpretations for classical algebraic systems, and I would like to know more about the geometry of algebraic rings, which I imagine as kind of like a bunch of slide rules near each other. But I still need to think about what GCD means in a slide rule interpretation of an algebraic system.
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