My hot(*) take is that teaching modular arithmetic before teaching things like ring homomorphisms and the universal property of quotients is kind of evil...

I think there are two kinds of people. The ones who think that there would be no way to motivate a quotient map of rings without an example like modular arithmetic. And then there are the ones who think that there would be no way to motivate and explain modular arithmetic without explaining what, precisely, it is doing in terms of universal properties (e.g. why some subtly different definition **must** be wrong).

I am both kinds of people, but if I must be honest, I am more the second kind of person. One important fact about comparing numbers "mod n" is that doing so forms a congruence for all the arithmetic operations. Students use this fact all the time, but they are using it ignorantly because they haven't been told that this is actually part of the definition, and they certainly haven't been equipped to deduce that it is OK from the weak intuitions we begrudgingly allow them to have.

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(*) it's called a hot take for a reason!!