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It seems people here are in a hurry to get a question closed as soon as it seems to deal with the same topic. I wonder how all 5 editors who voted to close Prove that every positive integer $n$ is a unique product of a square and a squarefree number missed the fact that I am asking for a proof by contradiction whereas the question that has been marked a duplicate of doesn't require proof by contradiction.

My question isn't answered yet and discussion on my question has stopped because it is closed. Is there a way I can reopen it?

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    $\begingroup$ Up again! Here! $\endgroup$ – user21436 May 2 '12 at 18:14
  • $\begingroup$ Great! Thanks @KannappanSampath. $\endgroup$ – Lone Learner May 2 '12 at 18:27
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Note that your question was reopened not because it asks for a proof by contradiction but, rather, because it asks for a proof using only gcds (vs. primes in the proposed duplicate). This is a proper distinction since GCD domains (domains where gcds exist) are more general than UFDs (domains with unique factorization into atoms = irreducibles). For example, there are GCD domains with no atoms, hence no primes, e.g. the ring of all algebraic integers, which has no atoms since every element factors $\rm\:a = \sqrt{a}^{\:\!2}.$

But at an elementary level, it is difficult (if not impossible) to classify proofs by contradiction (or not), because any proof can trivially be transformed into a "proof by contradiction" by first assuming the negation of the result desired, then giving the proof of the result. To classify proofs by contradiction (or not) requires technicalities in logic which cannot be easily presented at an elementary level.

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The question gets reopened when people vote to reopen it. Sometimes this happens without anyone taking any special action, but sometimes the way to do this is to do exactly what you did: post here on meta directing attention to it :)

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