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

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:
$\text{ " " } 12$
$\text{ " " / \}$
$\text{ " " 6 " 2}$
$\text{ " " /\ }$
$\text{ " " 2 3}$

12's prime factors are $2 , 2 , \mathmr{and} 3$

$\text{ " " } 9$
$\text{ " " / \}$
$\text{ " " 3 } 3$

9's prime factors are $3 \mathmr{and} 3$

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 $\textcolor{red}{2}$. Which number has more 2's: $12 \mathmr{and} 9$?
$12 : \textcolor{red}{\text{2,2}} , 3$
$9 : 3 , 3$
Obviously, 12 has more 2's because 9 has none.

Now what's the other number in our prime factorization? $\textcolor{b l u e}{3}$. Which number has more threes?
$12 : \textcolor{red}{\text{2,2}} , \cancel{3}$
$9 : \textcolor{b l u e}{\text{3,3}}$

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:
$\textcolor{red}{\text{2" xx color(red)2}}$ $\times \textcolor{b l u e}{3}$ $\times$ $\textcolor{b l u e}{3}$
$\textcolor{red}{4}$ $\times \textcolor{b l u e}{9} = \textcolor{p u r p \le}{36}$

36 is the least common multiple between 12 and 9.