Is LCM Associative? Tackling the Proof

[1MileCrash]

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..?

 

[micromass]

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.

 

[1MileCrash]

Alright, thank you.

I am currently trying an argument with prime factorization that seems... reasonable, but I will try this too.

 

[micromass]

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.

 

[1MileCrash]

Oh, I think I got you. They divide each other using that property (nearly) alone.