2-4 Prime Factorization

Course 2

Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson Presentation

Warm UpWrite each number as a product of two whole numbers in as many ways as possible.

1. 6

2. 16

3. 17

4. 36

5. 23

1 · 6, 2 · 3

1 · 16, 2 · 8, 4 · 4

1 · 17

Course 2

2-4 Prime Factorization

1 · 36, 2 · 18, 3 · 12, 4 · 9, 6 · 6

1 · 23

Problem of the Day

Nicholas bikes every third day and skates every other day. Today is April 5, and Nicholas biked and skated. On what date will he both bike and skate?April 11

Course 2

2-4 Prime Factorization

Learn to find the prime factorizations of composite numbers.

Course 2

2-4 Prime Factorization

Vocabulary

prime numbercomposite numberprime factorization

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Course 2

2-4 Prime Factorization

Course 2

2-4 Prime Factorization

In June 1999, Nayan Hajratwala discovered the first known prime number with more than one million digits. The new prime number, 26,972,593 – 1, has 2,098,960 digits.

A prime number is a whole number greater than 1 that has exactly two factors, 1 and itself. Three is a prime number because its only factors are 1 and 3.

Course 2

2-4 Prime Factorization

A composite number is a whole number that has more than two factors. Six is a composite number because it has more than two factors—1, 2, 3, and 6. The number 1 has exactly one factor and is neither prime nor composite.

A composite number can be written as the product of its prime factors. This is called the prime factorization of the number.

You can use a factor tree to find the prime factors of a composite number.

Write the prime factorization of the number.

Additional Example 1A: Using a Factor Tree to Find Prime Factorization

Course 2

2-4 Prime Factorization

A. 2424

8 · 3

4 · 2 · 3

2 · 2 · 2 · 3

Write 24 as the product oftwo factors.

Continue factoring until allfactors are prime.

The prime factorization of 24 is 2 · 2 · 2 · 3. Usingexponents, you can write this as 23 · 3.

Write the prime factorization of the number.

Additional Example B: Using a Factor Tree to Find Prime Factorization

Course 2

2-4 Prime Factorization

B. 150150

30 · 5

10 · 3 · 5

2 · 5 · 3 · 5

Write 150 as the productof two factors.

Continue factoring until all factors are prime.

The prime factorization of 150 is 2 · 3 · 5 · 5, or2 · 3 · 52.

Try This: Example 1A

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Course 2

2-4 Prime Factorization

Write the prime factorization of the number.

A. 3636

18 · 2

9 · 2 · 2

3 · 3 · 2 · 2

Write 36 as the product oftwo factors.

Continue factoring until allfactors are prime.

The prime factorization of 36 is 2 · 2 · 3 · 3. Usingexponents, you can write this as 22 · 32.

Try This: Example 1B

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Course 2

2-4 Prime Factorization

Write the prime factorization of the number.

B. 9090

45 · 2

9 · 5 · 2

3 · 3 · 5 · 2

Write 90 as the productof two factors.

Continue factoring until all factors are prime.

The prime factorization of 90 is 3 · 3 · 5 · 2, or2 · 32 · 5.

Course 2

2-4 Prime Factorization

You can also use a step diagram to find the prime factorization of a number. At each step, divide by the smallest possible prime number. Continue dividing until the quotient is 1. The prime factors are the number are the prime numbers you divided by.

Write the prime factorization of each number.

Additional Example 2A: Using a Step Diagram to Find Prime Factorization

Course 2

2-4 Prime Factorization

A. 476

476238119

171

22

717

Divide 476 by 2. Write the quotient below 476.

Keep dividing by a prime number.

Stop when the quotient is 1.

The prime factorization of 476 is 2 · 2 · 7 · 17, or22 · 7 · 17.

Write the prime factorization of the number.

Additional Example 2B: Using a Step Diagram to Find Prime Factorization

Course 2

2-4 Prime Factorization

B. 275

27555111

5511

Divide 275 by 5. Write the quotientbelow 275.

Stop when the quotient is 1.

The prime factorization of 275 is 5 · 5 · 11, or52 · 11.

Try This: Example 2A

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Course 2

2-4 Prime Factorization

Write the prime factorization of each number.

A. 324324

16281

27

1

22

33

Divide 324 by 2. Write the quotient below 324.

Keep dividing by a prime number.

Stop when the quotient is 1.

The prime factorization of 324 is 2 · 2 · 3 · 3 · 3 · 3, or22 · 34.

9333

Try This: Example 2B

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Course 2

2-4 Prime Factorization

Write the prime factorization of the number.

B. 325

32565131

5513

Divide 325 by 5. Write the quotientbelow 325.

Stop when the quotient is 1.

The prime factorization of 325 is 5 · 5 · 13, or52 · 13.

Course 2

2-4 Prime Factorization

There is only one prime factorization for any given composite number. Example 2A began by dividing 476 by 2, the smallest prime factor of 476. Beginning with any prime factor of 476 gives the same result.

476238119

171

22

717

4766834

171

72

217

The prime factorizations are 2 · 2 · 7 · 17 and7 · 2 · 2 · 17, which are the same as 17 · 2 · 2 · 7.

Lesson QuizUse a factor tree to find the prime factorization.

1. 27

2. 36

3. 28

Use a step diagram to find the prime

factorization.

4. 132

5. 52

6. 108

22 · 32

33

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22 · 7

22 · 3 · 11

Course 2

2-4 Prime Factorization

22 · 1322 · 33