1. © 2010 Pearson Prentice Hall. All rights reserved. CHAPTER 5 Number Theory and the Real Number System

2. © 2010 Pearson Prentice Hall. All rights reserved. 2 §5.1, Number Theory: Prime & Composite Numbers

3. © 2010 Pearson Prentice Hall. All rights reserved. Learning Targets I will determine divisibility. I will write the prime factorization of a composite number. I will find the greatest common divisor of two numbers. I will solve problems using the greatest common divisor. I will find the least common multiple of two numbers. I will solve problems using the least common multiple. 3

4. © 2010 Pearson Prentice Hall. All rights reserved. 4 Number Theory and Divisibility Number theory is primarily concerned with the properties of numbers used for counting, namely 1, 2, 3, 4, 5, and so on. The set of natural numbers is given by Natural numbers that are multiplied together are called the factors of the resulting product.

5. © 2010 Pearson Prentice Hall. All rights reserved. 5 Divisibility If a and b are natural numbers, a is divisible by b if the operation of dividing a by b leaves a remainder of 0. This is the same as saying that b is a divisor of a, or b divides a. This is symbolized by writing b|a. Example: We write 12|24 because 12 divides 24 or 24 divided by 12 leaves a remainder of 0. Thus, 24 is divisible by 12. Example: If we write 13|24, this means 13 divides 24 or 24 divided by 13 leaves a remainder of 0. But this is not true, thus, 13|24.

6. © 2010 Pearson Prentice Hall. All rights reserved. 6 A prime number is a natural number greater than 1 that has only itself and 1 as factors. A composite number is a natural number greater than 1 that is divisible by a number other than itself and 1. The Fundamental Theorem of Arithmetic Every composite number can be expressed as a product of prime numbers in one and only one way. One method used to find the prime factorization of a composite number is called a factor tree. Prime Factorization

7. © 2010 Pearson Prentice Hall. All rights reserved. 7 Example: Find the prime factorization of 700. Solution: Start with any two numbers whose product is 700, such as 7 and 100. Example: Prime Factorization using a Factor Tree Continue factoring the composite number, branching until the end of each branch contains a prime number.

8. © 2010 Pearson Prentice Hall. All rights reserved. 8 Thus, the prime factorization of 700 is 700 = 7  2  2  5  5 = 7  2 2  5 2 Notice, we rewrite the prime factorization using a dot to indicate multiplication, and arranging the factors from least to greatest. Example (continued)

9. © 2010 Pearson Prentice Hall. All rights reserved. 9 Greatest Common Divisor To find the greatest common divisor of two or more numbers ; 1.Write the prime factorization of each number. 2.Select each prime factor with the smallest exponent that is common to each of the prime factorizations. 3.Form the product of the numbers from step 2. The greatest common divisor is the product of these factors. Pairs of numbers that have 1 as their greatest common divisor are called relatively prime. For example, the greatest common divisor of 5 and 26 is 1. Thus, 5 and 26 are relatively prime.

10. © 2010 Pearson Prentice Hall. All rights reserved. 10 Example: Find the greatest common divisor of 216 and 234. Solution: Step 1. Write the prime factorization of each number. Example: Finding the Greatest Common Divisor

11. © 2010 Pearson Prentice Hall. All rights reserved. 11 216 = 2 3  3 3 234 = 2  3 2  13 Step 2. Select each prime factor with the smallest exponent that is common to each of the prime factorizations. Which exponent is appropriate for 2 and 3? We choose the smallest exponent; for 2 we take 2 1, for 3 we take 3 2. Example (continued)

12. © 2010 Pearson Prentice Hall. All rights reserved. 12 Step 3. Form the product of the numbers from step 2. The greatest common divisor is the product of these factors. Greatest common divisor = 2  3 2 = 2  9 = 18. Thus, the greatest common factor for 216 and 234 is 18. Example (continued)

13. © 2010 Pearson Prentice Hall. All rights reserved. 13 Least Common Multiple The least common multiple of two or more natural numbers is the smallest natural number that is divisible by all of the numbers. To find the least common multiple using prime factorization of two or more numbers: 1.Write the prime factorization of each number. 2.Select every prime factor that occurs, raised to the greatest power to which it occurs, in these factorizations. 3.Form the product of the numbers from step 2. The least common multiple is the product of these factors.

14. © 2010 Pearson Prentice Hall. All rights reserved. 14 Example: Find the least common multiple of 144 and 300. Solution: Step 1. Write the prime factorization of each number. 144 = 2 4  3 2 300 = 2 2  3  5 2 Step 2. Select every prime factor that occurs, raised to the greatest power to which it occurs, in these factorizations. 144 = 2 4  3 2 300 = 2 2  3  5 2 Example: Finding the Least Common Multiple

15. © 2010 Pearson Prentice Hall. All rights reserved. 15 Step 3. Form the product of the numbers from step 2. The least common multiple is the product of these factors. LCM = 2 4  3 2  5 2 = 16  9  25 = 3600 Hence, the LCM of 144 and 300 is 3600. Thus, the smallest natural number divisible by 144 and 300 is 3600. Example: continued

16. © 2010 Pearson Prentice Hall. All rights reserved. Critical Thinking: You and your brother both work the 4:00 p.m. to midnight shift at the movie theater. You have every sixth night off, while your brother has every tenth night off. Both of you were off on June 1. Your brother would like to see a movie with you. When will the two of you have the same night off again?

17. © 2010 Pearson Prentice Hall. All rights reserved. Homework: Page 236, #30 – 42 (e), 46 – 56 (e), 57 – 65.