If $a \mid bc$, then does $\frac{a}{(a,b)} \mid c$? I doubt anybody here is industrious enough to show this via a diagram, but who knows.
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$\begingroup$ I think almost every time I ask someone to draw a diagram they never know what to do or always complain that diagrams don't apply. $\endgroup$– Yosef QianCommented May 1, 2013 at 4:25
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$\begingroup$ The below diagram, as you can see, suffices to show the desired property. $\endgroup$– Yosef QianCommented May 1, 2013 at 4:30
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1 Answer
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Set $d=(a,b)$, $a=a'd, b=b'd$. We have $(a',b')=1$. Now, the hypothesis is $a|bc$, or $a'd|b'dc$. Cancelling $d$, we get $a'|b'c$. Since $(a',b')=1$, $a'|c$.
Diagram:
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$\begingroup$ I guess I was just saying that it might be interesting to develop a micro field of mathematics within a sub-field of mathematics that shows modular arithmetic and the properties of numbers with diagrams. $\endgroup$ Commented May 1, 2013 at 4:29