I am currently trying to self-study Stephen Abbott's Understanding Analysis. The first exercise asks to prove the irrationality of √3, and I understand the general idea of the contradiction by finding that the relatively prime integers p and q have a common factor. However, I am stuck on the idea that if p^2 is divisible by 3, then p is divisible by 3. Abbott's solution assumes this, but I have also seen proofs that analyze the situations where a and b are even or odd (such as NASA's). Even or odd really is just saying multiple of 2, which confuses me as to why the even/odd method (which is much less concise) would be used.
Sorry for the block of rambling text, I just want to start writing proofs the right way. I guess my real questions are:
If p^2 is divisible by a prime number, is p also divisible by that prime number? Can this just be assumed, or is there a theorem I have to mention in the proof? Why do some proofs analyze the even/odd situations of a and b? Are they more rigorous, and if they are not, why are they used, considering their added length and complexity? Finally, am I simply over thinking the idea of being rigorous and missing the big picture?