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I am a first-year maths student but I occasionally drift away from our taught material. Some years ago I saw the ZFC axioms for the first time, but now that I am in college, and although the stuff I've been taught so far is nowhere near ZFC (in terms of difficulty), it happened to me that we use the axiom of choice all the time in every module, even if we don't know it by name yet.
For example, in the proof that, for every non-negative integers $a, b$, there exist integers $q, r: a = bq + r$ (with the known restrictions on r), and the proof starts like this: $Choose$ the largest integer $q : qb <= a$... blah blah blah.
Is it the axiom of choice that allows us to execute this simple yet so important step?
And a couple more questions: Can you name some other simple proofs, theorems, results etc for which the axiom of choice is essential?
Also, I've read that AOC has long been a topic of dispute for mathematicians, and that even today, some people do not accept it. Are there any alternative axiomatic systems that work equally well without needing AOC? Thanks!