If $n\mid at$, show that $\frac{n}{\gcd(a,n)}\mid t$.
I've been trying to figure this out for hours, even though it seems very basic (I think I'm missing something fundamental here). What would be the way to show this?
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Anne Bauval
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Guy
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Does this answer your question? If $ a \mid bc $ then $\frac{a}{\gcd(a,b)} \mid c$? – Anne Bauval Jul 23 '23 at 16:02
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Let $d$ be the gcd of $a$ and $n$. Then $a=a_1d$ for some integer $a_1$, and $n=n_1d$ for some integer $n_1$, where $a_1$ and $n_1$ are relatively prime.
We are told that $n_1d$ divides $a_1dt$. It follows that $n_1$ divides $a_1t$. Since $n_1$ and $a_1$ are relatively prime, we conclude that $n_1$ divides $t$.
André Nicolas
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Great, thanks! The only part I don't understand would be, why would $a_{1}$ and $n_{1}$ (in $a=a_{1}d$, $n=n_{1}d$) be coprime? – Guy Sep 27 '15 at 20:58
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1If they are not coprime, suppose they have a common divisor $k\gt 1$. Then $kd$ is a common divisor of $a$ and $n$, contradicting the fact that $d$ is the greatest common divisor of $a$ and $n$. – André Nicolas Sep 27 '15 at 21:12
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You are welcome. Perhaps I should have filled in that detail in the answer. – André Nicolas Sep 27 '15 at 21:29
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