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LCM is associative

  • Thread starter 1MileCrash
  • Start date
  • #1
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45

Homework Statement



I need to prove that the least common multiple operation is associative.


Homework Equations





The Attempt at a Solution



Pages of crappy algebra trying to use the fact that LCM(a,b) = |ab|/gcd(a,b)

I hate to be "that guy" that doesn't post much of an attempt but I am getting nowhere with this. Maybe a hint or a fact about the LCM that will lead to a proof..?
 

Answers and Replies

  • #2
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3,277
Let ##x = \textrm{LCM}(a,\textrm{LCM}(b,c))## and ##y=\textrm{LCM}(\textrm{LCM}(a,b),c)##.

First, show that ##a## divides both ##x## and ##y##. And the same for ##b## and ##c##. Then show that ##\textrm{LCM}(b,c)## divides ##y## and that ##\textrm{LCM}(a,b)## divides ##x##.

Start with that.
 
  • #3
1,331
45
Alright, thank you.

I am currently trying an argument with prime factorization that seems... reasonable, but I will try this too.
 
  • #4
22,097
3,277
I'm trying to use the fact that if ##a## divides a number ##z## and if ##b## divides a number ##z##, then ##\textrm{LCM}(a,b)## divides ##z##. Do you know this fact? Try to prove it.
 
  • #5
1,331
45
Oh, I think I got you. They divide each other using that property (nearly) alone.
 

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