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Sure, to a mathematician, these things become old hat, but note that Apostols spends several pages developing properties of the GCD that aren't generally considered "obvious"; in particular, he argues for Theorem 1.2 (that any two integers a and b have a common divisor of the form ax + by, which is in turn therefore divisible by all their other common divisors) by tracing out the steps to recursively compute this combination-as-GCD (i.e., the Euclidean algorithm), recast as an inductive proof. I don't think the layperson would find this obvious at all (in the way they might have claimed FTA to be obvious), or even be aware of this fact, though it plays a key role in building up to the eventual proof of the FTA.



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