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3-1. Prime Factorization. Course 2. Warm Up Write 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. 1 · 36, 2 · 18, 3 · 12, 4 · 9, 6 · 6. 1 · 23. 3-1. Prime Factorization.

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Warm Up Write each number as a product of two whole numbers in as many ways as possible. 1. 6

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3-1

Prime Factorization

Course 2

Warm Up

Write 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

1 · 36, 2 · 18, 3 · 12, 4 · 9, 6 · 6

1 · 23

3-1

Prime Factorization

Course 2

Learn to find the prime factorizations of composite numbers.

3-1

Prime Factorization

Course 2

Insert Lesson Title Here

Vocabulary

prime number

composite number

prime factorization

3-1

Prime Factorization

Course 2

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 numberis 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.

3-1

Prime Factorization

Course 2

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.

3-1

Prime Factorization

Course 2

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

Write the prime factorization of the number.

A. 24

Write 24 as the product of

two factors.

24

8 · 3

Continue factoring until all

factors are prime.

4 · 2 · 3

2 · 2 · 2 · 3

The prime factorization of 24 is 2 · 2 · 2 · 3. Using

exponents, you can write this as 23 · 3.

3-1

Prime Factorization

Course 2

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

Write the prime factorization of the number.

B. 150

150

Write 150 as the product

of two factors.

30 · 5

Continue factoring until

all factors are prime.

10 · 3 · 5

2 · 5 · 3 · 5

The prime factorization of 150 is 2 · 3 · 5 · 5, or

2 · 3 · 52.

3-1

Prime Factorization

Course 2

Insert Lesson Title Here

Try This: Example 1A

Write the prime factorization of the number.

A. 36

Write 36 as the product of

two factors.

36

18 · 2

Continue factoring until all

factors are prime.

9 · 2 · 2

3 · 3 · 2 · 2

The prime factorization of 36 is 2 · 2 · 3 · 3. Using

exponents, you can write this as 22 · 32.

3-1

Prime Factorization

Course 2

Insert Lesson Title Here

Try This: Example 1B

Write the prime factorization of the number.

B. 90

90

Write 90 as the product

of two factors.

45 · 2

Continue factoring until

all factors are prime.

9 · 5 · 2

3 · 3 · 5 · 2

The prime factorization of 90 is 3 · 3 · 5 · 2, or

2 · 32 · 5.

3-1

Prime Factorization

Course 2

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.

3-1

Prime Factorization

Course 2

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

Write the prime factorization of each number.

A. 476

Divide 476 by 2. Write the

quotient below 476.

476

2

238

2

Keep dividing by a prime

number.

119

7

17

17

1

Stop when the quotient is 1.

The prime factorization of 476 is 2 · 2 · 7 · 17, or

22 · 7 · 17.

3-1

Prime Factorization

Course 2

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

Write the prime factorization of the number.

B. 275

Divide 275 by 5. Write the quotient

below 275.

275

5

55

5

11

11

Stop when the quotient is 1.

1

The prime factorization of 275 is 5 · 5 · 11, or

52 · 11.

3-1

Prime Factorization

Course 2

Insert Lesson Title Here

Try This: Example 2A

Write the prime factorization of each number.

A. 324

Divide 324 by 2. Write the

quotient below 324.

324

2

162

2

Keep dividing by a prime

number.

81

3

27

3

9

3

Stop when the quotient is 1.

3

3

1

The prime factorization of 324 is 2 · 2 · 3 · 3 · 3 · 3, or

22 · 34.

3-1

Prime Factorization

Course 2

Insert Lesson Title Here

Try This: Example 2B

Write the prime factorization of the number.

B. 325

Divide 325 by 5. Write the quotient

below 325.

325

5

65

5

13

13

Stop when the quotient is 1.

1

The prime factorization of 325 is 5 · 5 · 13, or

52 · 13.

3-1

Prime Factorization

Course 2

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.

476

476

2

7

238

68

2

2

119

34

7

2

17

17

17

17

1

1

The prime factorizations are 2 · 2 · 7 · 17 and

7 · 2 · 2 · 17, which are the same as 17 · 2 · 2 · 7.

3-1

Prime Factorization

Course 2

Insert Lesson Title Here

Lesson Quiz

Use 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

33

22 · 32

22 · 7

22 · 3 · 11

22 · 13

22 · 33