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

Question

Roman Alvarez · Accepted Answer

See process below:

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

Everly Taylor · Answer

undefined To find the least common multiple (LCM) using prime factorization, follow these steps:

1. Write the prime factorization of each number.
2. Identify all the unique prime factors from both sets of prime factorizations.
3. For each prime factor, take the highest power it appears with in either of the numbers' prime factorizations.
4. Multiply all these prime factors together to get the LCM.