The Language of Algebra

# 12 Prime Factorization and the Least Common Multiple

### Learning Objectives

By the end of this section, you will be able to:

• Find the prime factorization of a composite number
• Find the least common multiple (LCM) of two numbers

Before you get started, take this readiness quiz.

1. Is divisible by
If you missed this problem, review (Figure).
2. Is prime or composite?
If you missed this problem, review (Figure).
3. Write in exponential notation.
If you missed this problem, review (Figure).

### 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 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.

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 as you work through this section.

#### 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 We can start with any factor pair such as and We write and below with branches connecting them.

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

The factor is prime, so we circle it. The factor is composite, and it factors into We write these factors under the Since is prime, we circle both

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

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

Note that we could have started our factor tree with any factor pair of We chose and but the same result would have been the same if we had started with and and

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.

Find the prime factorization of 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 2s and the 3 since they are prime. Now all of the branches end in a prime. Write the product of the circled numbers. Write in exponential form.

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

Find the prime factorization using the factor tree method:

2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 5, or 24 ⋅ 5

Find the prime factorization using the factor tree method:

2 ⋅ 2 ⋅ 3 ⋅ 5, or 22 ⋅ 3 ⋅ 5

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

Solution
 We start with the factor pair 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.

Draw a factor tree of

Find the prime factorization using the factor tree method:

2 ⋅ 3 ⋅ 3 ⋅ 7, or 2 ⋅ 32 ⋅ 7

Find the prime factorization using the factor tree method:

2 ⋅ 3 ⋅ 7 ⋅ 7, or 2 ⋅ 3 ⋅ 72

#### 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 we divide by the smallest prime factor of

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

Then we divide by the next prime; so we divide by

We continue dividing up the ladder in this way until the quotient is prime. Since the quotient, 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.

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

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.

Find the prime factorization of 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.

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

Find the prime factorization using the ladder method:

2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 5, or 24 ⋅ 5

Find the prime factorization using the ladder method:

2 ⋅ 2 ⋅ 3 ⋅ 5, or 22 ⋅ 3 ⋅ 5

Find the prime factorization of 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.

Find the prime factorization using the ladder method.

2 ⋅ 3 ⋅ 3 ⋅ 7, or 2 ⋅ 32 ⋅ 7

Find the prime factorization using the ladder method.

2 ⋅ 3 ⋅ 7 ⋅ 7, or 2 ⋅ 3 ⋅ 72

### 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 and 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.

We see that and appear in both lists. They are common multiples of and 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 and is

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.

Find the LCM of and by listing multiples.

Solution

List the first several multiples of and of Identify the first common multiple.

The smallest number to appear on both lists is so is the least common multiple of and

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

Find the least common multiple (LCM) of the given numbers:

36

Find the least common multiple (LCM) of the given numbers:

72

#### 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 and

We start by finding the prime factorization of each number.

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

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

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

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.

Find the LCM of and 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. The LCM of 15 and 18 is 90.

Find the LCM using the prime factors method.

60

Find the LCM using the prime factors method.

105

Find the LCM of and 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. The LCM of 50 and 100 is 100.

Find the LCM using the prime factors method:

440

Find the LCM using the prime factors method:

360

### Key Concepts

• 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.
• 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.
• Find the LCM 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.
• 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.

### Section Exercises

#### Practice Makes Perfect

Find the Prime Factorization of a Composite Number

In the following exercises, find the prime factorization of each number using the factor tree method.

2 ⋅ 43

2 ⋅ 2 ⋅ 3 ⋅ 11

3 ⋅ 3 ⋅ 7 ⋅ 11

5 ⋅ 23

3 ⋅ 3 ⋅ 5 ⋅ 5 ⋅ 11

1560

In the following exercises, find the prime factorization of each number using the ladder method.

2 ⋅ 2 ⋅ 2 ⋅ 7

2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 7

17 ⋅ 23

2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3

2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 5

In the following exercises, find the prime factorization of each number using any method.

2 ⋅ 3 ⋅ 5 ⋅ 5

3 ⋅ 5 ⋅ 5 ⋅ 7

2 ⋅ 2 ⋅ 3 ⋅ 3

2 ⋅ 5 ⋅ 5 ⋅ 7

Find the Least Common Multiple (LCM) of Two Numbers

In the following exercises, find the least common multiple (LCM) by listing multiples.

24

30

120

300

In the following exercises, find the least common multiple (LCM) by using the prime factors method.

24

120

420

In the following exercises, find the least common multiple (LCM) using any method.

42

120

#### Everyday Math

Grocery shopping Hot dogs are sold in packages of ten, but hot dog buns come in packs of eight. What is the smallest number of hot dogs and buns that can be purchased if you want to have the same number of hot dogs and buns? (Hint: it is the LCM!)

40

Grocery shopping Paper plates are sold in packages of and party cups come in packs of What is the smallest number of plates and cups you can purchase if you want to have the same number of each? (Hint: it is the LCM!)

#### Writing Exercises

Do you prefer to find the prime factorization of a composite number by using the factor tree method or the ladder method? Why?

Do you prefer to find the LCM by listing multiples or by using the prime factors method? Why?

#### Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

### Chapter Review Exercises

#### Use the Language of Algebra

Use Variables and Algebraic Symbols

In the following exercises, translate from algebra to English.

the product of 3 and 8

the quotient of 24 and 6

50 is greater than or equal to 47

The sum of n and 4 is equal to 13

Identify Expressions and Equations

In the following exercises, determine if each is an expression or equation.

equation

expression

Simplify Expressions with Exponents

In the following exercises, write in exponential form.

23

x6

In the following exercises, write in expanded form.

8 ⋅ 8 ⋅ 8 ⋅ 8

yyyyy

In the following exercises, simplify each expression.

81

128

Simplify Expressions Using the Order of Operations

In the following exercises, simplify.

20

18

74

31

#### Evaluate, Simplify, and Translate Expressions

Evaluate an Expression

In the following exercises, evaluate the following expressions.

58

when

26

Identify Terms, Coefficients and Like Terms

In the following exercises, identify the terms in each expression.

12n2,3n, 1

In the following exercises, identify the coefficient of each term.

6

In the following exercises, identify the like terms.

3, 4, and 3x, x

Simplify Expressions by Combining Like Terms

In the following exercises, simplify the following expressions by combining like terms.

24a

14x

12n + 11

10y2 + 2y + 3

Translate English Phrases to Algebraic Expressions

In the following exercises, translate the following phrases into algebraic expressions.

the difference of and

x − 6

the sum of and twice

the product of and

3n ⋅ 9

the quotient of and

times the sum of and

5(y + 1)

less than the product of and

Jack bought a sandwich and a coffee. The cost of the sandwich was more than the cost of the coffee. Call the cost of the coffee Write an expression for the cost of the sandwich.

c + 3

The number of poetry books on Brianna’s bookshelf is less than twice the number of novels. Call the number of novels Write an expression for the number of poetry books.

#### Solve Equations Using the Subtraction and Addition Properties of Equality

Determine Whether a Number is a Solution of an Equation

In the following exercises, determine whether each number is a solution to the equation.

1. yes
2. no

1. yes
2. no

1. no
2. yes

Model the Subtraction Property of Equality

In the following exercises, write the equation modeled by the envelopes and counters and then solve the equation using the subtraction property of equality.

x + 3 = 5; x = 2

Solve Equations using the Subtraction Property of Equality

In the following exercises, solve each equation using the subtraction property of equality.

6

11

Solve Equations using the Addition Property of Equality

In the following exercises, solve each equation using the addition property of equality.

23

34

Translate English Sentences to Algebraic Equations

In the following exercises, translate each English sentence into an algebraic equation.

The sum of and is equal to

7 + 33 = 44

The difference of and is equal to

The product of and is equal to

4 ⋅ 8 = 32

The quotient of and is equal to

Twice the difference of and gives

2(n − 3) = 76

The sum of five times and is

Translate to an Equation and Solve

In the following exercises, translate each English sentence into an algebraic equation and then solve it.

Eight more than is equal to

x + 8 = 35; x = 27

less than is

The difference of and is

q − 18 = 57; q = 75

The sum of and is

Mixed Practice

In the following exercises, solve each equation.

h = 42

z = 33

q = 8

v = 56

#### Find Multiples and Factors

Identify Multiples of Numbers

In the following exercises, list all the multiples less than for each of the following.

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48

8, 16, 24, 32, 40, 48

Use Common Divisibility Tests

In the following exercises, using the divisibility tests, determine whether each number is divisible by

2, 3, 6

2, 3, 5, 6, 10

Find All the Factors of a Number

In the following exercises, find all the factors of each number.

1, 2, 3, 5, 6, 10, 15, 30

1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180

Identify Prime and Composite Numbers

In the following exercises, identify each number as prime or composite.

prime

composite

#### Prime Factorization and the Least Common Multiple

Find the Prime Factorization of a Composite Number

In the following exercises, find the prime factorization of each number.

2 ⋅ 2 ⋅ 3 ⋅ 7

2 ⋅ 5 ⋅ 5 ⋅ 7

Find the Least Common Multiple of Two Numbers

In the following exercises, find the least common multiple of each pair of numbers.

45

175

#### Everyday Math

Describe how you have used two topics from The Language of Algebra chapter in your life outside of your math class during the past month.

### Chapter Practice Test

In the following exercises, translate from an algebraic equation to English phrases.

fifteen minus x

In the following exercises, identify each as an expression or equation.

equation

1. Write in exponential form.
2. Write in expanded form and then simplify.
1. n6
2. 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 = 243

In the following exercises, simplify, using the order of operations.

36

5

45

In the following exercises, evaluate each expression.

125

36

Simplify by combining like terms.

In the following exercises, translate each phrase into an algebraic expression.

more than

x + 5

the quotient of and

three times the difference of

3(ab)

Caroline has fewer earrings on her left ear than on her right ear. Call the number of earrings on her right ear, Write an expression for the number of earrings on her left ear.

In the following exercises, solve each equation.

n = 31

In the following exercises, translate each English sentence into an algebraic equation and then solve it.

less than is

y − 15 = 32; y = 47

the sum of and is

List all the multiples of that are less than

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48

Find all the factors of

Find the prime factorization of

23 ⋅ 33 ⋅ 5

Find the LCM (Least Common Multiple) of and

### Glossary

least common multiple
The smallest number that is a multiple of two numbers is called the least common multiple (LCM).
prime factorization
The prime factorization of a number is the product of prime numbers that equals the number.