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**Factors and Prime Factorization**

2.2 Factors and Prime Factorization

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**Finding the Factors of Numbers**

To perform many operations, it is necessary to be able to factor a number. Since 7 · 9 = 63, both 7 and 9 are factors of 63, and 7 · 9 is called a factorization of 63. Objective A 2

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**Finding Factors of Numbers**

Definition Example Factors: When numbers are multiplied to form a product, each number is called a factor. The different factorizations of 6 are: So the factors of 6 are: 1, 2, 3 and 6. Practice Problems 1 Find all the factors of each of the numbers. Objective A P 122

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**Prime and Composite Numbers**

Prime Numbers A prime number is a natural number that has exactly two different factors 1 and itself. Composite Numbers A composite number is any natural number, other than 1, that is not prime. Objective A pp 4

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Examples Determine whether each number is prime or composite. Explain your answers. a. 16 b. 31 c. 49 Composite, it has more than two factors: 1, 2, 4, 8, 16. Prime, its only factors are 1 and 31. Objective A Composite, it has more than two factors: 1, 7, 49. 5

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**Identifying Prime and Composite Numbers**

Definition Example Prime Number: A natural number that has exactly two different factors, 1 and itself. The first several prime numbers are: 2, 3, 5, 7, 11, 13, 17 Composite Number: If a natural number other than 1 is not a prime number, it is called a composite number. The number 10 has more than two factors: 1, 2, 5, and 10 Practice Problems 2 Determine whether each number is Prime or Composite. Objective B p 123

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**Prime Factorization Prime Factorization**

The prime factorization of a number is the factorization in which all the factors are prime numbers. Every whole number greater than 1 has exactly one prime factorization. Objective A p 123 7

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**Examples Find the prime factorization of 63.**

The first prime number 2 does not divide evenly, but 3 does. Because 21 is not prime, we divide again. The quotient 7 is prime, so we are finished. The prime factorization of 63 is 3 · 3 · 7. Objective A 8

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**Finding Prime Factorizations**

Definition Example Prime Factorization: The factorization in which all the factors are prime numbers. The prime factorization for 84. because 42 is not a prime number we must divide it by a prime number. because 21 is not a prime number we must divide it by a prime number. because 7 is a prime number we can now write the prime factorization of 84. Objective C p 124

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Factor Trees Another way to find the prime factorization is to use a factor tree. Objective A 10

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**Finding Prime Factorizations**

Definition Practice Problem 3 Prime Factorization: The factorization in which all the factors are prime numbers. 28 2 14 2 7 Objective C p 125

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**Examples Find the prime factorization of 30.**

Write 30 as the product of two numbers. Continue until all factors are prime. 30 • 3 • • 5 The prime factorization of 30 is 2 · 3 · 5. Objective A 12

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**Examples Find the prime factorization of 36.**

Write 36 as the product of two numbers. Continue until all factors are prime. 36 • 3 • • 2 The prime factorization of 36 is 3 · 3 · 2 · 2 or 32 · 22. Objective A 13

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**Finding Prime Factorizations**

Definition Practice Problem 4 Prime Factorization: The factorization in which all the factors are prime numbers. 120 2 60 2 30 2 15 Objective C 3 5 p 124

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**Finding Prime Factorizations**

Definition Practice Problem 5 Prime Factorization: The factorization in which all the factors are prime numbers. 756 126 6 2 3 2 63 3 21 3 7 Objective C p 125

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**Finding Prime Factorizations**

Definition Practice Problem 6 Prime Factorization: The factorization in which all the factors are prime numbers. 70 2 35 5 7 Objective C p 125

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**Finding Prime Factorizations**

Definition Practice Problem 7a Prime Factorization: The factorization in which all the factors are prime numbers. 30 2 15 5 3 Objective C p 126

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**Finding Prime Factorizations**

Definition Practice Problem 7b Prime Factorization: The factorization in which all the factors are prime numbers. 56 2 28 2 14 2 7 Objective C Problem 7c: 72 and Problem 8: 117 p 126

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DONE

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Divisibility Tests Objective A 20

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