# LCM is associative

## Homework Statement

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

## 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..?

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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 $x$.

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.