How can you find the least common multiple using prime factorization?

Let's come up with a problem so that I can show you the process.

What is the least common multiple of 12 and 9?

Let's prime factor each of the numbers:
`" " " " 12`
`" " " / \"`
`" " " 6 " 2"`
`" " " /\ "`
`" " " 2 3"`

12's prime factors are `2,2, and 3`

`" " " " 9`
`" " " / \"`
`" " " 3 " 3`

9's prime factors are `3 and 3`

Now make a chart with both of the numbers:
`12: 2,2,3`
`9: 3,3`

This is where it gets a little tricky. What we're going to is find the lowest number in our prime factorization. That number is `color(red)2`. Which number has more 2's: `12 or 9`?
`12: color(red)"2,2",3`
`9: 3,3`
Obviously, 12 has more 2's because 9 has none.

Now what's the other number in our prime factorization? `color(blue)3`. Which number has more threes?
`12: color(red)"2,2",cancel3`
`9: color(blue)"3,3"`

9 has more 3's than 12, so I am going to cross out the other 3. We only want the part with the most threes.

Put all of the highlighted numbers down into one multiplication problem:
`color(red)"2" xx color(red)2"` `xx color(blue)3` `xx` `color(blue)3`
`color(red)4` `xx color(blue)9 = color(purple)36`

36 is the least common multiple between 12 and 9.

This is a really helpful video on YouTube about this topic:least common multiple