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LCM is associative

  1. Mar 23, 2014 #1
    1. The problem statement, all variables and given/known data

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


    2. Relevant equations



    3. 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..?
     
  2. jcsd
  3. Mar 23, 2014 #2

    micromass

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    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.
     
  4. Mar 23, 2014 #3
    Alright, thank you.

    I am currently trying an argument with prime factorization that seems... reasonable, but I will try this too.
     
  5. Mar 23, 2014 #4

    micromass

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    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.
     
  6. Mar 23, 2014 #5
    Oh, I think I got you. They divide each other using that property (nearly) alone.
     
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