1. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Chapter 2 Fractions

2. 2-1-2 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Section 2.1 Factors, Prime Factorizations, and Least Common Multiples

3. 2-1-3 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Factors of a Natural Number Rule for Finding Factors of a Natural Number Divide the natural number by each of the numbers 1, 2, 3, and so on. If the natural number is divisible by one of these numbers, then both the divisor and the quotient are factors of the natural number. Continue until the factors begin to repeat.

4. 2-1-4 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Rules for Divisibility A NUMBER IS DIVISIBLE BY IF 2The number is even. 3The sum of the digits is divisible by 3. 4The number named by the last two digits is divisible by 4. 5The last digit is either 0 or 5.

5. 2-1-5 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Rules for Divisibility A NUMBER IS DIVISIBLE BY IF 6The number is even and the sum of the digits is divisible by 3. 8The number named by the last three digits is divisible by 8. 9The sum of digits is divisible by 9. 10The last digit is 0.

6. 2-1-6 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Example Find the factors of each number. a.64 b.60

7. 2-1-7 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Solution Strategy a. Divide 64 by 1, 2, 3, and so on. 64 ÷ 1 = 641 and 64 are factors. 64 ÷ 2 = 322 and 32 are factors. 64 ÷ 3Does not divide evenly. 64 ÷ 4 = 164 and 16 are factors. 64 ÷ 5Does not divide evenly. 64 ÷ 6Does not divide evenly. 64 ÷ 7 Does not divide evenly. 64 ÷ 8 = 88 and 8 are factors.

8. 2-1-8 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Solution Strategy a. (continued) 64 ÷ 10Does not divide evenly. 64 ÷ 12Does not divide evenly. 64 ÷ 13Does not divide evenly. 64 ÷ 14Does not divide evenly. 64 ÷ 15Does not divide evenly. 64 ÷ 16 = 416 and 4 are factors. Repeat factors-Stop! 64 ÷ 9Does not divide evenly.

9. 2-1-9 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Solution Strategy b. Divide 60 by 1, 2, 3, and so on. 60 ÷ 1 = 601 and 60 are factors. 60 ÷ 2 = 302 and 30 are factors. 60 ÷ 3 = 203 and 20 are factors. 60 ÷ 4 = 154 and 15 are factors. 60 ÷ 5 = 125 and 12 are factors. 60 ÷ 6 = 106 and 10 are factors. 60 ÷ 7 Does not divide evenly.

10. 2-1-10 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Solution Strategy b. (continued) 60 ÷ 8 = 10Does not divide evenly. 60 ÷ 9 Does not divide evenly. 60 ÷ 10 10 and 6 are factors. Repeat factors-Stop!

11. 2-1-11 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Prime and Composite Numbers Prime A natural number greater than 1 that has only two factors (divisors), namely, 1 and itself. Composite A natural number greater than 1 that has more than two factors (divisors).

12. 2-1-12 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Prime Numbers Less Than 100 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

13. 2-1-13 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Example Determine whether each number is prime, composite, or neither. a.16b. 13 c. 23 d. 72e. 0 f. 19

14. 2-1-14 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Solution Strategy a.16compositeFactors: 1, 2, 4, 8, and 16 b.13primeFactors: 1 and itself c. 23primeFactors: 1 and itself d.72compositeFactors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72 e.0neitherBy definition f. 19primeFactors: 1 and itself

15. 2-1-15 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Factoring a Number To factor a number means to express the number as a product of factors. As an example, consider the number 18. There are three two-number factorizations.

16. 2-1-16 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Prime Factoring a Number A prime factorization of a whole number is a factorization in which each factor is prime. The prime factorization of 18 is

17. 2-1-17 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Factor Trees A factor tree is an illustration used to determine the prime factorization of a composite number. or

18. 2-1-18 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Example Find the prime factorization of each number. a. 60b. 90c. 288

19. 2-1-19 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Solution Strategy a. 60 The prime factorization of 60 is 2 2 3 5 = 2 2 3 5. 60 10 5 6 223

20. 2-1-20 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Solution Strategy b. 90 The prime factorization of 90 is 2 3 3 5 = 2 3 2 5.

21. 2-1-21 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Solution Strategy c. 288

22. 2-1-22 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Solution Strategy c. 288 The prime factorization of 288 is 2 2 2 2 2 3 3 = 2 5 3 2.

23. 2-1-23 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Least Common Multiple A multiple of a number is the product of the number and any natural number. A common multiple is a multiple that is shared by a set of two or more natural numbers. A least common multiple or LCM is the smallest multiple shared by a set of two or more numbers.

24. 2-1-24 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Finding the Least Common Multiple

25. 2-1-25 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Example Find the LCM of 4, 6, and 10 using prime factorization.

26. 2-1-26 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Solution Strategy 4 = 2 2 6 = 2 3 10 = 2 5 LCM = 2 2 3 5 = 60 Find the LCM of 4, 6, and 10 using prime factorization.

27. 2-1-27 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Alternative Method of Finding the LCM

28. 2-1-28 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Example Find the LCM of 4, 6, and 10 using the alternative method.

29. 2-1-29 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Solution Strategy LCM = 2 2 3 5 = 60 Find the LCM of 4, 6, and 10 using the alternative method.

30. 2-1-30 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Example Apply your knowledge You have volunteered to be the “barbeque chef ” for a school party. From experience, you know that shrimp should be turned every 2 minutes and salmon should be turned every 3 minutes. How often will they be turned at the same time?

31. 2-1-31 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. Solution Strategy LCM = 2 3 = 6 2 = 1 2 3 = 1 3 The shrimp and salmon will be turned at the same time every 6 minutes.