Prime Factorization and the Least Common Multiple

2.5 Prime Factorization and the Least Common Multiple

The topics covered in this section are:

2.5.1 Find the Prime Factorization of a Composite Number

In the previous section, we found the factors of a number. Prime numbers have only two factors, the number $1$ and the prime number itself. Composite numbers have more than two factors, and every composite number can be written as a unique product of primes. This is called the prime factorization of a number. When we write the prime factorization of a number, we are rewriting the number as a product of primes. Finding the prime factorization of a composite number will help you later in this course.

PRIME FACTORIZATION

The prime factorization of a number is the product of prime numbers that equals the number.

MANIPULATIVE MATHEMATICS

Doing the Manipulative Mathematics activity “Prime Numbers” will help you develop a better sense of prime numbers.

You may want to refer to the following list of prime numbers less than $50$ as you work through this section.

$2,3,5,7,11,13,17,19,23,29,31,37,41,43,47$

Prime Factorization Using the Factor Tree Method

One way to find the prime factorization of a number is to make a factor tree. We start by writing the number, and then writing it as the product of two factors. We write the factors below the number and connect them to the number with a small line segment—a “branch” of the factor tree.

If a factor is prime, we circle it (like a bud on a tree), and do not factor that “branch” any further. If a factor is not prime, we repeat this process, writing it as the product of two factors and adding new branches to the tree.

We continue until all the branches end with a prime. When the factor tree is complete, the circled primes give us the prime factorization.

For example, let’s find the prime factorization of $36$. We can start with any factor pair such as $3$ and $12$. We write $3$ and $12$ below $36$ with branches connecting them.

The figure shows a factor tree with the number 36 at the top. Two branches are splitting out from under 36. The right branch has a number 3 at the end with a circle around it. The left branch has the number 12 at the end.

The factor $3$ is prime, so we circle it. The factor $12$ is composite, so we need to find its factors. Let’s use $3$ and $4$. We write these factors on the tree under the $12$.

The figure shows a factor tree with the number 36 at the top. Two branches are splitting out from under 36. The right branch has a number 3 at the end with a circle around it. The left branch has the number 12 at the end. Two more branches are splitting out from under 12. The right branch has the number 4 at the end and the left branch has the number 3 at the end.

The factor $3$ is prime, so we circle it. The factor $4$ is composite, and it factors into $2 \cdot 2$. We write these factors under the $4$. Since $2$ is prime, we circle both $2$s.

The figure shows a factor tree with the number 36 at the top. Two branches are splitting out from under 36. The right branch has a number 3 at the end with a circle around it. The left branch has the number 12 at the end. Two more branches are splitting out from under 12. The right branch has the number 4 at the end and the left branch has the number 3 at the end with a circle around it. Two more branches are splitting out from under 4. Both the left and right branch have the number 2 at the end with a circle around it.

The prime factorization is the product of the circled primes. We generally write the prime factorization in order from least to greatest.

$2 \cdot 2 \cdot 3\cdot 3$

In cases like this, where some of the prime factors are repeated, we can write prime factorization in exponential form.

$2 \cdot 2 \cdot 3\cdot 3$

$2^{2} \cdot 3^{2}$

Note that we could have started our factor tree with any factor pair of $36$. We chose $12$ and $3$, but the same result would have been the same if we had started with $2$ and $18,4$ and $9$, or $6$ and $6$.

HOW TO: Find the prime factorization of a composite number using the tree method.

  1. Find any factor pair of the given number, and use these numbers to create two branches.
  2. If a factor is prime, that branch is complete. Circle the prime.
  3. If a factor is not prime, write it as the product of a factor pair and continue the process.
  4. Write the composite number as the product of all the circled primes.

Example 1

Find the prime factorization of $48$ using the factor tree method.

Solution

We can start our tree using any factor pair of $48$. Let’s use $2$ and $24$.
We circle the $2$ because it is prime and so that branch is complete.
.
Now we will factor $24$. Let’s use $4$ and $6$..
Neither factor is prime, so we do not circle either.
We factor the $4$, using $2$ and $2$.
We factor $6$, using $2$ and $3$.

We circle the $2$s and the $3$ since they are prime. Now all of the branches end in a prime.
.
Write the product of the circled numbers.$2 \cdot 2 \cdot 2 \cdot 2 \cdot 3$
Write in exponential form$2^{4} \cdot 3$

Check this on your own by multiplying all the factors together. The result should be $48$.

Example 2

Find the prime factorization of $84$ using the factor tree method.

Solution

We start with the factor pair of $4$ and $21$.
Neither factor is prime so we factor them further.
.
Now the factors are all prime, so we circle them..
Then we write $84$ as the product of all circled primes.$2 \cdot 2 \cdot 3 \cdot 7$
$2^{2} \cdot 3 \cdot 7$

Draw a factor tree of 84.

Prime Factorization Using the Ladder Method

The ladder method is another way to find the prime factors of a composite number. It leads to the same result as the factor tree method. Some people prefer the ladder method to the factor tree method, and vice versa.

To begin building the “ladder,” divide the given number by its smallest prime factor. For example, to start the ladder for $36$ we divide $36$ by $2$, the smallest prime factor of $36$.

The image shows the division of 2 into 36 to get the quotient 18. This division is represented using a division bracket with 2 on the outside left of the bracket, 36 inside the bracket and 18 above the 36, outside the bracket.

To add a “step” to the ladder, we continue dividing by the same prime until it no longer divides evenly.

The image shows the division of 2 into 36 to get the quotient 18. This division is represented using a division bracket with 2 on the outside left of the bracket, 36 inside the bracket and 18 above the 36, outside the bracket. Another division bracket is written around the 18 with a 2 on the outside left of the bracket and a 9 above the 18, outside of the bracket.

Then we divide by the next prime; so we divide $9$ by $3$.

The image shows the division of 2 into 36 to get the quotient 18. This division is represented using a division bracket with 2 on the outside left of the bracket, 36 inside the bracket and 18 above the 36, outside the bracket. Another division bracket is written around the 18 with a 2 on the outside left of the bracket and a 9 above the 18, outside of the bracket. Another division bracket is written around the 9 with a 3 on the outside left of the bracket and a 3 above the 9, outside of the bracket.

We continue dividing up the ladder in this way until the quotient is prime. Since the quotient, $3$, is prime, we stop here.

Do you see why the ladder method is sometimes called stacked division?

The prime factorization is the product of all the primes on the sides and top of the ladder.

$2 \cdot 2 \cdot 3 \cdot 3$

$2^{2} \cdot 3^{2}$

Notice that the result is the same as we obtained with the factor tree method.

HOW TO: Find the prime factorization of a composite number using the ladder method.

  1. Divide the number by the smallest prime.
  2. Continue dividing by that prime until it no longer divides evenly.
  3. Divide by the next prime until it no longer divides evenly.
  4. Continue until the quotient is a prime.
  5. Write the composite number as the product of all the primes on the sides and top of the ladder.

Example 3

Find the prime factorization of $120$ using the ladder method.

Solution

Divide the number by the smallest prime, which is $2$.
Continue dividing by $2$ until it no longer divides evenly..
Divide by the next prime, $3$..
The quotient, $5$, is prime, so the ladder is complete. Write the prime factorization of $120$..$2 \cdot 2 \cdot 2 \cdot 3 \cdot 5$
$2^{3} \cdot 3 \cdot 5$

Check this yourself by multiplying the factors. The result should be $120$.

Example 4

Find the prime factorization of $48$ using the ladder method.

Solution

Divide the number by the smallest prime, $2$..
Continue dividing by $2$ until it no longer divides evenly..
The quotient, $3$, is prime, so the ladder is complete. Write the prime factorization of $48$.$2 \cdot 2 \cdot 2 \cdot 2\cdot 3$
$2^{4} \cdot 3$

2.5.2 Find the Least Common Multiple (LCM) of Two Numbers

One of the reasons we look at multiples and primes is to use these techniques to find the least common multiple of two numbers. This will be useful when we add and subtract fractions with different denominators.

Listing Multiples Method

A common multiple of two numbers is a number that is a multiple of both numbers. Suppose we want to find common multiples of $10$ and $25$. We can list the first several multiples of each number. Then we look for multiples that are common to both lists—these are the common multiples.

$10:10,20,30,40, \mathbf{50},60,70,80,90, \mathbf{100}, 110,…$

$25:25, \mathbf{50}, 75, \mathbf{100}, 125, …$

We see that $50$ and $100$ appear in both lists. They are common multiples of $100$ and $25$. We would find more common multiples if we continued the list of multiples for each.

The smallest number that is a multiple of two numbers is called the least common multiple (LCM). So the least LCM of $10$ and $25$ is $50$.

HOW TO: Find the least common multiple (LCM) of two numbers by listing multiples.

  1. List the first several multiples of each number.
  2. Look for multiples common to both lists. If there are no common multiples in the lists, write out additional multiples for each number.
  3. Look for the smallest number that is common to both lists.
  4. This number is the LCM.

Example 5

Find the LCM of $15$ and $20$ by listing multiples.

Solution

List the first several multiples of $15$ and of $20$. Identify the first common multiple.

$15:15,30,45, \mathbf{60}, 75,90,105,120$

$20:20,40, \mathbf{60}, 80,100,12,140,160$

The smallest number to appear on both lists is $60$, so $60$ is the least common multiple of $15$ and $20$.

Notice that $120$ is on both lists, too. It is a common multiple, but it is not the least common multiple.

Prime Factors Method

Another way to find the least common multiple of two numbers is to use their prime factors. We’ll use this method to find the LCM of $12$ and $18$.

We start by finding the prime factorization of each number.

$12=2 \cdot 2 \cdot 3$

$18 = 2 \cdot 3 \cdot 3$

Then we write each number as a product of primes, matching primes vertically when possible.

$12=2 \cdot 2 \cdot 3$

$18 = 2 \cdot 3 \cdot 3$

Now we bring down the primes in each column. The LCM is the product of these factors.

The image shows the prime factorization of 12 written as the equation 12 equals 2 times 2 times 3. Below this equation is another showing the prime factorization of 18 written as the equation 18 equals 2 times 3 times 3. The two equations line up vertically at the equal symbol. The first 2 in the prime factorization of 12 aligns with the 2 in the prime factorization of 18. Under the second 2 in the prime factorization of 12 is a gap in the prime factorization of 18. Under the 3 in the prime factorization of 12 is the first 3 in the prime factorization of 18. The second 3 in the prime factorization has no factors above it from the prime factorization of 12. A horizontal line is drawn under the prime factorization of 18. Below this line is the equation LCM equal to 2 times 2 times 3 times 3. Arrows are drawn down vertically from the prime factorization of 12 through the prime factorization of 18 ending at the LCM equation. The first arrow starts at the first 2 in the prime factorization of 12 and continues down through the 2 in the prime factorization of 18. Ending with the first 2 in the LCM. The second arrow starts at the next 2 in the prime factorization of 12 and continues down through the gap in the prime factorization of 18. Ending with the second 2 in the LCM. The third arrow starts at the 3 in the prime factorization of 12 and continues down through the first 3 in the prime factorization of 18. Ending with the first 3 in the LCM. The last arrow starts at the second 3 in the prime factorization of 18 and points down to the second 3 in the LCM.

Notice that the prime factors of $12$ and the prime factors of $18$ are included in the LCM. By matching up the common primes, each common prime factor is used only once. This ensures that $36$ is the least common multiple.

HOW TO: Find the LCM using the prime factors method.

  1. Find the prime factorization of each number.
  2. Write each number as a product of primes, matching primes vertically when possible.
  3. Bring down the primes in each column.
  4. Multiply the factors to get the LCM.

Example 6

Find the LCM of $15$ and $18$ using the prime factors method.

Solution

Write each number as a product of primes..
Write each number as a product of primes, matching primes vertically when possible..
Bring down the primes in each column..
Multiply the factors to get the LCM.LCM $=2 \cdot 3 \cdot 3 \cdot5$
The LCM of $15$ and $18$ is $90$.

Example 7

Find the LCM of $50$ and $100$ using the prime factors method.

Solution

Write the prime factorization of each number..
Write each number as a product of primes, matching primes vertically when possible..
Bring down the primes in each column..
Multiply the factors to get the LCM.LCM $=2 \cdot 2 \cdot 5 \cdot 5$
The LCM of $50$ and $100$ is $100$.
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