# LCM is associative

1. Mar 23, 2014

### 1MileCrash

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. Mar 23, 2014

### micromass

Staff Emeritus
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 $x$.

3. Mar 23, 2014

### 1MileCrash

Alright, thank you.

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

4. Mar 23, 2014

### micromass

Staff Emeritus
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. Mar 23, 2014

### 1MileCrash

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