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Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 2.2 Factors and Prime Factorization.

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1 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 2.2 Factors and Prime Factorization

2 Martin-Gay, Basic Mathematics, 4e 22 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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.

3 Martin-Gay, Basic Mathematics, 4e 33 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. DefinitionExample 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. P 122 Practice Problems 1 Find all the factors of each of the numbers. Finding Factors of Numbers

4 Martin-Gay, Basic Mathematics, 4e 44 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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. pp

5 Martin-Gay, Basic Mathematics, 4e 55 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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. Composite, it has more than two factors: 1, 7, 49.

6 Martin-Gay, Basic Mathematics, 4e 66 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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 p 123 Practice Problems 2 Determine whether each number is Prime or Composite. Identifying Prime and Composite Numbers

7 Martin-Gay, Basic Mathematics, 4e 77 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Prime Factorization Every whole number greater than 1 has exactly one prime factorization. Prime Factorization The prime factorization of a number is the factorization in which all the factors are prime numbers. p 123

8 Martin-Gay, Basic Mathematics, 4e 88 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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.

9 Martin-Gay, Basic Mathematics, 4e 99 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. DefinitionExample 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. p 124 Finding Prime Factorizations

10 Martin-Gay, Basic Mathematics, 4e 10 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Factor Trees Another way to find the prime factorization is to use a factor tree.

11 Martin-Gay, Basic Mathematics, 4e 11 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Definition Practice Problem 3 Prime Factorization: The factorization in which all the factors are prime numbers. p Finding Prime Factorizations

12 Martin-Gay, Basic Mathematics, 4e 12 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Examples Find the prime factorization of 30. Write 30 as the product of two numbers. Continue until all factors are prime The prime factorization of 30 is 2 · 3 · 5.

13 Martin-Gay, Basic Mathematics, 4e 13 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Examples Find the prime factorization of 36. Write 36 as the product of two numbers. Continue until all factors are prime The prime factorization of 36 is 3 · 3 · 2 · 2 or 3 2 · 2 2.

14 Martin-Gay, Basic Mathematics, 4e 14 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. DefinitionPractice Problem 4 Prime Factorization: The factorization in which all the factors are prime numbers. p Finding Prime Factorizations

15 Martin-Gay, Basic Mathematics, 4e 15 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. DefinitionPractice Problem 5 Prime Factorization: The factorization in which all the factors are prime numbers. p Finding Prime Factorizations

16 Martin-Gay, Basic Mathematics, 4e 16 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. DefinitionPractice Problem 6 Prime Factorization: The factorization in which all the factors are prime numbers. p Finding Prime Factorizations

17 Martin-Gay, Basic Mathematics, 4e 17 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. DefinitionPractice Problem 7a Prime Factorization: The factorization in which all the factors are prime numbers. p Finding Prime Factorizations

18 Martin-Gay, Basic Mathematics, 4e 18 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. DefinitionPractice Problem 7b Prime Factorization: The factorization in which all the factors are prime numbers. p Problem 7c: 72 and Problem 8: 117 Finding Prime Factorizations

19 Martin-Gay, Basic Mathematics, 4e 19 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. DONE

20 Martin-Gay, Basic Mathematics, 4e 20 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Divisibility Tests


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