If $\gcd(a,bc)=d$, we can say that $d|a$ and $d|bc$ but what about $d|b$, $d|c$ individually?

Asked Jul 01 '23 at 11:31

Active Jul 01 '23 at 12:46

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I am learning about the proof of the theorem: If $(b,c)=1$, then $(a,b)(a,c)=(a,bc)$. I have completed more than half of the theorem proof but at the end I let $(a,bc)=d$, and i know that $d \mid a,d \mid bc$ can i write also $d \mid c$ and $d \mid b$ individually?. Because $bc$ is the product of two integers. Please clarify my confusion. Thank you.

edited Jul 01 '23 at 12:18

Bill Dubuque

272,048

asked Jul 01 '23 at 11:31

Muhammad Seyab

7

Just try examples. You should easily see that we need not have $d$ dividing either $b$ or $c$. – lulu Jul 01 '23 at 11:33
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To get modern notation for divisibility in MathJax use the character | or \mid. – jjagmath Jul 01 '23 at 11:45
The text book i m using writing directly the that $d/d_1c$ for another integer $d_1$. I wanna know that how can he write $d/d_1c$ even we dont know about $d/c$. – Muhammad Seyab Jul 01 '23 at 11:53
Only if $d$ is a prime (or $1$) , we have $d\mid bc\implies d\mid b\ or\ d\mid c$ – Peter Jul 01 '23 at 11:59
Your assertion is incorrect. You can try a stronger proof. – Itoz Darbien Jul 01 '23 at 12:04
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No you can't write until and unless d is prime or one. If d is not prime then consider following counterexample: Take d=6,b=2 and c=3. Then $d \mid ab$ that is $6|2×3$ but 6 doesn't divide 2 or 3. – Afzal Jul 01 '23 at 12:12
Ok thank you for your replies i got the point. – Muhammad Seyab Jul 01 '23 at 12:14

If $\gcd(a,bc)=d$, we can say that $d|a$ and $d|bc$ but what about $d|b$, $d|c$ individually?

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