I tried using Bézout's theorem: if $(a,b) = 1$, there are $m,n$ such that $am+bn=1$, $amc+bnc=c$, but it doesn't work, I don't get good result... I know that $a\mid c²$ and $b\mid c²$.
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Here's a MathJax tutorial :) – Shaun Jan 23 '20 at 22:11
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3This looks like a job for the Fundamental Theorem of Arithmetic. – Shaun Jan 23 '20 at 22:12
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Hint:
Consider the prime decomposition of $c$: $\;c=\displaystyle\prod_{1\le i\le n} p_i^{r_i}$, so that $$c^2=p_1^{2r_1}\dotsm p_n^{2r_n}$$
Each primary factor $p_i^{2r_i}$ divides the product $ab$, but as $a$ and $b$ are coprime, it can only divide one of them, by Euclid's lemma. Can you continue to show that there is a partition in the set of primary factors of $c^2$: those which divide $a$ and those which divide $b$, and conclude?
Bernard
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oh thanks, man. I have a question, how did you put the symbol of product? and they exponents in "p"? I'm new here... – Kayan Tchian Jan 23 '20 at 22:54
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1Just as in LaTeX:
$\displaystyle \prod_{1\le i \le n} p_{i}^{r_i}$. There must links here to tutorials for the language used, which isMathJax(not as powerful as LaTeX, though). – Bernard Jan 23 '20 at 23:03