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

1 Answer
Mar 11, 2018

See process below:

Explanation:

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 12
" " " / \" / \
" " " 6 " 2" 6 2
" " " /\ " /\
" " " 2 3" 2 3

12's prime factors are 2,2, and 32,2,and3

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

9's prime factors are 3 and 33and3

Now make a chart with both of the numbers:
12: 2,2,312:2,2,3
9: 3,39:3,3

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)22. Which number has more 2's: 12 or 912or9?
12: color(red)"2,2",312:2,2,3
9: 3,39:3,3
Obviously, 12 has more 2's because 9 has none.

Now what's the other number in our prime factorization? color(blue)33. 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