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[Update: I see that the equality is wrong: Assume that p divide the right hand side. If p|(c,e) and p|(d,e), then p need not to divide the left hand side. Thank you all for your comments]

Let $a,b,c,d,e$ be integers. I am digging the following equality:

$\Big[ (a,b) , (a,c,e), (b,d,e) \Big] = \Big( \big[a, (d,e) \big] , \big[b,(c,e) \big] \Big) $

where $[\ldots]$ and $(\ldots)$ denote LCM and GCD respectively. I can show that this equality holds as follows: take a prime $p$ that divide the left side, then it will also divide the right side, and vice versa. But is there another elegant way to show that this equality holds, for instance, by using the properties of GCD and LCM?

Bill Dubuque
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John Watts
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    The proof you gave is wrong. You'll need to prove that a given prime have the same exponent on both sides, not only that they are divisible by the same primes. – jjagmath Apr 13 '21 at 10:23
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    In the RHS, the outer LCM has only one element. Is that a typo? – jjagmath Apr 13 '21 at 10:28
  • Thanks for the remark, the question is edited. – John Watts Apr 13 '21 at 10:34
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    The equality is false, so look for a counterexample. – jjagmath Apr 13 '21 at 10:41
  • Actually even the proof you suggest would not work (it is possible that prime $p$ divides right hand side but does not divide left hand side) – Sil Apr 13 '21 at 10:53
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    I see that the equality is wrong: Assume that $p$ divide the right hand side. If $p \vert (c,e)$ and $p \vert (d,e)$, then $p$ need not to divide the left hand side. Thank you all for your comments. – John Watts Apr 13 '21 at 11:07

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