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