Prime Factorization and Least Common Multiple | Grade 5 Math | LevelKite

You can already tell a prime number from a composite number, and you can find the factor pairs of a number. Now you will put those skills together.

You can split a composite number into smaller factors. Keep splitting, and you reach numbers you cannot split any further: primes. The prime factorization of a number is that number written as a product of primes. That means only prime numbers, multiplied together. For example, $12 = 2 \times 2 \times 3$ . Each of those factors is prime, and together they multiply to make $12$ .

Think of primes as building blocks. In this lesson you will find the building blocks of a number. Then you will use them to find a multiple that two numbers share.

A factor tree is a diagram that shows how you split a number, one step at a time. You build it like this:

Write the composite number at the top.
Split it into a factor pair. Draw two branches down, one to each factor. Skip the pair that uses $1$ , because it does not break the number into smaller parts.
Look at the number at the end of each branch. If it is prime, circle it. That branch is finished.
If it is composite, split it again in the same way.
Stop when every branch ends in a circled prime.

Each split keeps the value the same, because the two factors multiply to make the number above them. So when the tree is finished, the circled primes multiply to make the number at the top. That product is the prime factorization.

Worked example

A factor tree for 36

Use a factor tree to write $36$ as a product of prime numbers.

Write 36 at the top. Choose a factor pair of 36 that does not use 1. Here the pair is 4 and 9. Draw one branch down to 4 and one branch down to 9. The split works because 4 times 9 is 36.
$36 = 4 \times 9$
Look at the 4. It is composite, so split it. Its factor pair without 1 is 2 and 2. Draw two branches down from the 4, one to each 2. Both 2s are prime, so circle them.
$4 = 2 \times 2$
Look at the 9. It is composite too, so split it. Its factor pair without 1 is 3 and 3. Draw two branches down from the 9, one to each 3. Both 3s are prime, so circle them.
$9 = 3 \times 3$
Every branch now ends in a circled prime, so the tree is finished. Write 36 again, with 2 times 2 in place of the 4 and 3 times 3 in place of the 9. Check by multiplying: 2 times 2 is 4, 3 times 3 is 9, and 4 times 9 is 36.
$36 = 2 \times 2 \times 3 \times 3$

Answer

The prime factorization of $36$ is $2 \times 2 \times 3 \times 3$ .

That tree started with $4 \times 9$ . But $36$ has other factor pairs too, such as $6 \times 6$ and $2 \times 18$ . Does a different start lead to different primes?

The table shows three factor trees for $36$ . Each row is one tree. Read a row from left to right, the same way you follow the branches down a tree. Watch where each row ends.

Three factor trees for 36

First split	Later splits	Primes at the ends, multiplied
$36 = 4 \times 9$	$4 = 2 \times 2$ and $9 = 3 \times 3$	$2 \times 2 \times 3 \times 3 = 36$
$36 = 6 \times 6$	$6 = 2 \times 3$ and $6 = 2 \times 3$	$2 \times 3 \times 2 \times 3 = 36$
$36 = 2 \times 18$	$18 = 2 \times 9$ , then $9 = 3 \times 3$	$2 \times 2 \times 3 \times 3 = 36$

Different trees, same primes. Every tree ends in two $2$ s and two $3$ s. The order can change, but the order does not matter when you multiply. So $36$ has just one prime factorization: $36 = 2 \times 2 \times 3 \times 3$ .

This works for every composite number. Start with any factor pair you like. As long as you keep splitting until every branch ends in a prime, you reach the same primes.

Now think about multiples. A common multiple of two numbers is a number that is a multiple of both. For example, $12$ is a multiple of $4$ , because $4 \times 3 = 12$ . It is also a multiple of $6$ , because $6 \times 2 = 12$ . So $12$ is a common multiple of $4$ and $6$ . So are $24$ and $36$ .

The least common multiple, or LCM, is the smallest common multiple. For $4$ and $6$ , the LCM is $12$ .

Here is one way to find an LCM. List the multiples of each number in order. The first number that lands in both lists is the LCM.

Worked example

Find an LCM by listing multiples

Find the least common multiple of $6$ and $8$ by listing multiples.

List the multiples of 6 in order. Multiply 6 by 1, then by 2, then by 3, and keep going. Write down the first few.
$6, 12, 18, 24, 30, \ldots$
List the multiples of 8 the same way. After each new number, look for it in the list for 6. Is 8 there? No. Is 16 there? No. Is 24 there? Yes, so you can stop.
$8, 16, 24, \ldots$
24 is the first number in both lists, so it is the smallest multiple that 6 and 8 share. Match it to the multiplication: 24 is 6 times 4, and 24 is also 8 times 3.
$6 \times 4 = 24 \quad \text{and} \quad 8 \times 3 = 24$

Answer

The least common multiple of $6$ and $8$ is $24$ .

That shared $2$ is a clue. Prime factors can show you exactly what an LCM needs.

A multiple of $6$ is $6$ times some whole number, so it always has the primes of $6$ inside it: a $2$ and a $3$ . In the same way, a multiple of $8$ always has the three $2$ s of $8$ inside it. The LCM is the smallest number that holds both sets of primes.

So build it. Start with the three $2$ s that $8$ needs. One of them also does the job for $6$ . Now $6$ is only missing its $3$ . Put in one $3$ and you are done: $2 \times 2 \times 2 \times 3 = 24$ . That is the same LCM your lists gave. The table below lines up the counts.

Building the LCM of 6 and 8: keep the larger count of each prime

Number	Prime factorization	How many 2s	How many 3s
$6$	$2 \times 3$	one	one
$8$	$2 \times 2 \times 2$	three	none
The LCM, $24$	$2 \times 2 \times 2 \times 3$	three	one

Key concept

Primes are building blocks

Every composite number breaks down into one set of primes, no matter which factor tree you draw. Those primes show what every multiple of the number must have inside it.

The least common multiple of two numbers is the smallest number that holds all the prime factors of both. To build it, take each prime in turn. Count how many times it appears in each prime factorization, and keep the larger count. If the counts match, keep that count. Then multiply the primes you kept.

Worked example

Find an LCM with prime factorizations

Use prime factorizations to find the least common multiple of $12$ and $18$ . Then check your answer by listing multiples.

Find the prime factorization of each number with a factor tree. 12 splits into 2 and 6, and 6 splits into 2 and 3. 18 splits into 2 and 9, and 9 splits into 3 and 3. Every branch now ends in a prime.
$12 = 2 \times 2 \times 3 \quad \text{and} \quad 18 = 2 \times 3 \times 3$
Compare the 2s. 12 has two 2s and 18 has one 2. Keep the larger count, two. A multiple of 12 needs both 2s, and those two 2s also cover the one 2 that 18 needs.
$\text{Keep two 2s: } 2 \times 2$
Compare the 3s. 12 has one 3 and 18 has two 3s. Keep the larger count, two. A multiple of 18 needs both 3s, and those two 3s also cover the one 3 that 12 needs.
$\text{Keep two 3s: } 3 \times 3$
Multiply the primes you kept to build the LCM. 2 times 2 is 4, 3 times 3 is 9, and 4 times 9 is 36.
$2 \times 2 \times 3 \times 3 = 4 \times 9 = 36$
Check by listing multiples of each number until one appears in both lists. Multiples of 12: 12, 24, 36. Multiples of 18: 18, 36. The first shared multiple is 36, so the two methods agree.
$12 \times 3 = 36 \quad \text{and} \quad 18 \times 2 = 36$

Answer

The least common multiple of $12$ and $18$ is $36$ . Listing multiples gives the same answer.

Here is the big idea to take with you. Primes are the building blocks of numbers. A factor tree breaks a composite number into its prime factorization, and every tree for that number ends in the same primes.

The least common multiple of two numbers is the smallest multiple they share. You can find it by listing multiples until the lists first match. You can also build it from prime factors by keeping the larger count of each prime. Both ways give the same answer, so use one to check the other.