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CHAPTER 5

CHAPTER 5. Number Theory and the Real Number System. §5.1, Number Theory: Prime & Composite Numbers. 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.

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CHAPTER 5

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  1. CHAPTER 5 Number Theory and the Real Number System

  2. §5.1, Number Theory: Prime & Composite Numbers

  3. 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.

  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. 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. Prime Factorization • 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.

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

  8. Example (continued) Thus, the prime factorization of 700 is 700 = 7  2  2  5  5 = 7  22 52 Notice, we rewrite the prime factorization using a dot to indicate multiplication, and arranging the factors from least to greatest.

  9. Greatest Common Divisor To find the greatest common divisor of two or more numbers; • Write the prime factorization of each number. • Select each prime factor with the smallest exponent that is common to each of the prime factorizations. • 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. Example: Finding the Greatest Common Divisor Example: Find the greatest common divisor of 216 and 234. Solution:Step 1. Write the prime factorization of each number.

  11. Example (continued) 216 = 23 33 234 = 2  32 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 21, for 3 we take 32.

  12. Example (continued) 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  32 = 2  9 = 18. Thus, the greatest common factor for 216 and 234 is 18.

  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: • Write the prime factorization of each number. • Select every prime factor that occurs, raised to the greatest power to which it occurs, in these factorizations. • Form the product of the numbers from step 2. The least common multiple is the product of these factors.

  14. Example: Finding the Least Common Multiple Example: Find the least common multiple of 144 and 300. Solution:Step 1. Write the prime factorization of each number. 144 = 24 32 300 = 22 3  52 Step 2. Select every prime factor that occurs, raised to the greatest power to which it occurs, in these factorizations. 144 = 24 32 300 = 22 3  52

  15. Example: continued Step 3. Form the product of the numbers from step 2. The least common multiple is the product of these factors. LCM = 24 32 52 = 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.

  16. 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. Homework: Page 236, #30 – 42 (e), 46 – 56 (e), 57 – 65.

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