Section 2.1/2.2 An Introduction to Number Theory/Integers. The counting numbers or natural numbers are N = {1, 2, 3, }.

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1 Section 2.1/2.2 An Introduction to Number Theory/Integers The counting numbers or natural numbers are N = {1, 2, 3, }. A natural number n is called the product of the natural numbers a and b if a b = n. The numbers a and b are called the factors of n. I If a b = n, we know that n a = b and n b = a, therefore a and b are also called divisors of n. For example, the factors of 12 are 1, 2, 3, 4, 6, 12. The divisors of 8 are 1, 2, 4, 8. Every natural number greater than 1 can be classified as either a prime number or a composite number. A prime number is a natural number greater than 1 that has exactly two factors (or divisors), itself and 1. For example, the first ten primes are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 The number 2 is the only even prime number. A composite number is a natural number that is divisible by a number other than itself and 1. For example, 4, 6, 8, 9, 10, 12 are composite numbers. The number 1 is neither prime nor composite; it is called a unit. The Fundamental Theorem of Arithmetic Every composite number can be expressed as a unique product of prime numbers. This process is called prime factorization. Section 2.1/2.2 An Introduction to Number Theory/Integers 1

2 Rules of Divisibility Using rules of divisibility can help in determining whether a number is prime or can speed up the process in finding the prime factorization of a number. How to determine whether a number is prime? For large numbers, this is a difficult quesiton. But for small numbers n, list all known primes smaller than n, and check if any are divisors of n. Example 1: Determine whether 119 is prime. Section 2.1/2.2 An Introduction to Number Theory/Integers 2

3 Example 2: Write 196 as a product of primes. Example 3: Write 6000 as a product of primes. Section 2.1/2.2 An Introduction to Number Theory/Integers 3

4 Greatest Common Divisor The greatest common divisor (GCD) of a set of natural numbers is the largest natural number that divides (without remainder) every number in that set. To Find the Greatest Common Divisor of Two or More Numbers 1. Determine the prime factorization of each number. 2. Find each prime factor with the smallest exponent that appears in each of the prime factorizations. 3. Determine the product of the factors found in step 2. Example 4: Find the GCD of a. 54 and 90 b. 225 and 525 c. 9 and 14 9 and 14 are called relatively prime since the GCD is 1. Section 2.1/2.2 An Introduction to Number Theory/Integers 4

5 Least Common Multiple The least common multiple (LCM) of a set of natural numbers is the smallest natural number that is divisible by each element in the set. To Find the Least Common Multiple of Two or More Numbers 1. Determine the prime factorization of each number. 2. List each prime factor with the greatest exponent that appears in any of the prime factorizations. 3. Determine the product of the factors found in step 2. Example 5: Find the LCM of a. 20 and 36 b. 18, 78 and 198 Section 2.1/2.2 An Introduction to Number Theory/Integers 5

6 Example 6: Hotdogs are sold in packages of 8, and hotdog buns are sold in packages of 10. Henry is having a cookout and wants to grill hotdogs and have the number of hotdogs and buns be the same. What's the smallest number of hotdogs he should buy so that he needs only full packages of both hotdogs and buns. Example 7: Sarah is the program director at a retreat center. She is purchasing new tables for the dining room, and is trying to decide which size table to buy. The tables she is considering seat up to 12 people. The retreat center typically hosts weekend retreats for either 72, 80 or 120 people depending on the group. Sarah wants to be able to divide each group evenly into separate tables. Furthermore, she would like those table groups to be as large as possible for better dinner conversation. How many people should Sarah seat at each table? Section 2.1/2.2 An Introduction to Number Theory/Integers 6

7 The integers are the numbers in the set: Z = {,-4, -3, -2, -1, 0, 1, 2, 3, 4, }. Example 8: Evaluate the expression. a b c. ( 8 + 6) 20 5 ( 3) d e Example 9: If the temperature is 10 degrees below zero at 6PM, and drops another 5 degrees by 8PM, what is the temperature at 8PM? Example 10: If the temperature is 7 degrees above zero at noon and then drops 10 degrees by 9PM, what is the temperature at 9PM? Section 2.1/2.2 An Introduction to Number Theory/Integers 7

8 Enter your answer choices by the deadline in your CASA account under EMCF named Popper In order to convert the Hindu-Arabic numeral 5000 to a Babylonian numeral, we must divide by which of the following values first? A. 1 B. 60 C D E. 2. Which would be the correct expanded form in Hindu-Arabic of the given Mayan numeral? 4 60 A. 4(1) + 8(18 20) B. 8(1) + 4(18 20) C. 2 8(18 20) + 4(18 20 ) D. 8(1) + 4(20) E. 0(1) + 8(20) + 4(18 20) 3. Practice Tests are for extra credit. A. True B. False 4. Which is the correct Roman numeral for 40? A. XXXX B. XL C. XXXIIIIIIIIII D. VVL 5. Give the Roman numeral for 49. A. XLIX B. VCIV C. XXXXIX D. IL Section 2.1/2.2 An Introduction to Number Theory/Integers 8

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