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I am trying to prove that for every integer $n \ge 1$, there exists uniquely determined $a > 0$ and $b > 0$ such that $n = a^2 b$, where $b$ is squarefree.

I am trying to prove this using the properties of divisibility and GCD only. Is it possible?

Let me assume that $n = a^2 b = a'^2b'$ where $a \ne a'$ and $b \ne b$'. Can we show a contradiction now?

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    The question this has been dupped to did not include the added condition of using only properties of divisibility and of the GDC—indeed, the answers there do not satisfy this condition.2012-05-02
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    Please reopen this question. This question is not a duplicate. This question requires proof by contradiction which I can't find in the link provided as "possible duplicate".2012-05-02
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    It is probably worth mentioning that the question to which the previous comments refer is [Show that every n can be written uniquely in the form n=ab, with a square-free and b a perfect square](http://math.stackexchange.com/questions/21282/show-that-every-n-can-be-written-uniquely-in-the-form-n-ab-with-a-squa). (See [revision history](http://math.stackexchange.com/posts/139737/revisions).)2015-06-12

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