How to prove that if $c$ divides $ab$ and $\operatorname{gcd}(a,c)=1$, then show that $c$ divides $b$.
that means if $c|ab$ and $(a,c)=1 \implies c|b$.
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Presumably you mean $(a,c) = 1$. – ah11950 Apr 07 '14 at 11:21
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Does this answer your question? If $\gcd(a,b)= 1$ and $a$ divides $bc$ then $a$ divides $c\ $ [Euclid's Lemma] – Anne Bauval Sep 22 '23 at 15:45
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Hint. If $\gcd(a,c)=1$ then there exist integers $x,y$ such that $$ax+cy=1\ .$$ Now multiply both sides by $b$ and explain why $c$ is a factor of the left hand side.
David
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