Prove that $a$ divides $bc$ involving $\gcd$

Asked Apr 26 '23 at 08:48

Active Apr 26 '23 at 16:35

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Show that $a|bc$ if and only if $\frac{a}{\gcd(a,b)}|c$.

For the $\Rightarrow$ direction we know that there is a $u$ so that $bc=ua$ and we need to show the existence of a $v$ so that $c=v\frac{a}{\gcd(a,b)}$. From what we know $c=\frac{ua}{b}$. Choosing $v=\frac{u \gcd(a,b)}{b}$ leads to the desired result. A similar argument can be made for the opposite direction.

Is this correct? Because it seems too simple for me as it doesn't involve properties of the $\gcd$ at all.

edited Apr 26 '23 at 16:35

Bill Dubuque

282,220

asked Apr 26 '23 at 08:48

illuminatitruthseeker

1

You can easily compare your proof with the ones from a duplicate. Then you can replace your vague statement "A similar argument leads to..." by a valid proof. "Is this correct?" This depends on the grading scheme. It is certainly not yet full points. – Dietrich Burde Apr 26 '23 at 08:53
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you need to make sure each of these numbers is a positive integer. – Suraj Apr 26 '23 at 08:54
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... in particular $v$ – Henry Apr 26 '23 at 09:09
Use $x\mid y$ for $x\mid y$. – Shaun Apr 26 '23 at 09:44

Prove that $a$ divides $bc$ involving $\gcd$

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