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Slide 1 / 69 Slide 2 / 69 Table of Contents · Prime and Composite Numbers · Prime Factorization Whole Numbers · Common Factors · Greatest Common Factor · Relatively Prime · Least Common Multiple Slide 3 / 69 Slide 4 / 69 Prime and Composite Numbers Slide 5 / 69 1 The smallest prime number is _______. Slide 6 / 69 2 49 is not a prime number. True False Slide 7 / 69 3 This list contains 3 prime numbers: 1, 2, 3, 5, 9, and 12 Slide 8 / 69 4 This list contains 3 prime numbers: 5, 9, 20, 31, 42, 53, and 63 True True False False Slide 9 / 69 5 This list contains 3 prime numbers: 5, 9, 20, 31, 42, 53, and 63 Slide 10 / 69 6 This list contains 3 prime numbers: 15, 19, 23, 37, 47, 55, and 63 True True False False Slide 11 / 69 7 This list contains 3 prime numbers: 25, 29, 33, 38, 45, 57, and 76 Slide 12 / 69 The Sieve of Erastosenes True Find the prime numbers by sifting out the multiples of each prime. False Example: 2 is prime. Multiples of 2: 2, 4, 6, 8, 10, 12, 14... How do we know that the multiples of 2 are not prime? Slide 13 / 69 Slide 14 / 69 The Sieve of Erastosenes A Composite Number can be divided evenly by numbers other than 1 or itself. Sift out the multiples of each prime. Examples: What are you left with? 1 is NOT composite. Why not? X Slide 15 / 69 Is 18 prime or composite? Explain Slide 16 / 69 Is 63 prime or composite? Explain 18 is composite because it can be divided evenly by more than 1 and itself. 18 can be evenly divided by: 1, 2, 3, 6, 9, and 18. 63 is composite because it can be divided evenly by more than 1 and itself. 63 can be evenly divided by: 1, 3, 7, 9, 21, and 63. Slide 17 / 69 8 43 is _________ Slide 18 / 69 9 30 is _________ A Prime A Prime B Composite B Composite Slide 19 / 69 Slide 20 / 69 10 33 is _________ A Prime B Composite Factoring a Number Slide 21 / 69 Slide 22 / 69 Factors Factors are the numbers you multiply together to get another number. Example: 3 and 6 are factors of 18, because 3 x 6 = 18. Also, 2 x 9 =18, so 2 and 9 are also factors of 18. What are two other factors of 18? Prime Factorization Slide 23 / 69 Slide 24 / 69 Process for factoring a number into primes is the process of factoring a number so that all of the factors are prime numbers. What is the prime factorization of 18? 1. Divide the given number by the smallest prime number possible. 2. Continue to divide by the smallest prime number possible. 3. Keep dividing until the quotient (answer) is one. Example: 12 = 2 x 2 x 3 = 22 x 3 2 18 18 = 2 x 3 x 3 3 9 3 3 1 2 12 2 6 3 3 1 click for = 2 x 32 answer Slide 25 / 69 What is the prime factorization of 24? 2 24 24 = 2 x 2 x 2 x 3 2 12 click = 2 x 3 for answer 3 2 6 3 3 Slide 26 / 69 11 What is the prime factorization of 30? A 2x3x5 B 6x5 C 5x6 D 2 x 15 1 Slide 27 / 69 12 What is the prime factorization of 24? Slide 28 / 69 13 What is the prime factorization of 45? A 3x8 A 3 x 15 B 2x2x6 B 32 x 5 C 23 x 3 C 9x 5 D 2x2x2x3 D 52 x 3 Slide 29 / 69 14 What is the prime factorization of 60? Slide 30 / 69 15 What is the prime factorization of 100? A 2 x 3 x 10 A 2 x 3 x 10 B 2x5x2x3 B 2x5x2x3 2 C 2 x3x5 C 22 x 3 x 5 D 22 x 15 D 22 x 15 Slide 31 / 69 Common Factors A common factor is a number that is a factor of two or more numbers. Find the common factors of 12 and 16. Factors of 12: 1, 2, 3, 4, 6, click for12answer Factors of 16: 1, 2, 4,click 8, 16 for answer Common factors: 1, 2, 4 click for answer Slide 32 / 69 Common Factors Find the common factors of 18 and 24. Factors of 18: 1, 2, 3,click 6, 9,for18answer Factors of 24: 1, 2, 3,click 4, 6,for8,12, 24 answer Common factors: 1, 2, 3, 4, 6 click for answer What is the Greatest Common Factor? What is the Greatest Common Factor? Greatest Common Factor: 4 Greatest Common Factor: 6 click for answer Slide 33 / 69 16 The greatest common factor for 12 and 48 is ____. Slide 34 / 69 17 The greatest common factor for 24 and 36 is ____. A 2 A 2 B 4 B 4 C 6 C 6 D 12 D 12 Slide 35 / 69 18 The greatest common factor for 42 and 64 is ____. click for answer Slide 36 / 69 19 The greatest common factor for 50 and 100 is ____. A 2 A 5 B 4 B 10 C 6 C 25 D 8 D 50 Slide 37 / 69 20 The greatest common factor for 36 and 90 is ____. A 3 B 9 C 12 D 18 Slide 38 / 69 Greatest Common Factor We can use prime factorization to find the greatest common factor (GCF). 1. Factor the given numbers into primes. 2. Circle the factors that are common. 3. Multiply the common factors together to find the common factor. Slide 39 / 69 Slide 40 / 69 Slide 41 / 69 Slide 42 / 69 greatest 21 Use prime factorization to find the GCF of 18 and 44. Slide 43 / 69 22 Use prime factorization to find the GCF of 28 and 70. Slide 44 / 69 23 Use prime factorization to find the GCF of 55 and 110. Slide 45 / 69 24 Use prime factorization to find the GCF of 52 and 78. Slide 46 / 69 25 Use prime factorization to find the GCF of 72 and 75. Slide 47 / 69 Relatively Prime: Two or more numbers are relatively prime if their greatest common factor is 1. Example: 15 and 32 are relatively prime because their GCF is 1. Name two numbers that are relatively prime. Slide 48 / 69 26 Identify at least two numbers that are relatively prime to 9. A 16 B 15 C 28 D 36 Slide 49 / 69 27 7 and 35 are not relatively prime. Slide 50 / 69 28 Name a number that is relatively prime to 20. True False Slide 51 / 69 29 Name a number that is relatively prime to 5 and 18. Slide 53 / 69 Slide 52 / 69 30 Find two numbers that are relatively prime A 7 B 14 C 15 D 49 Slide 54 / 69 A multiple of a whole number is the product of the number and any nonzero whole number. Least Common Multiple A multiple that is shared by two or more numbers is a common multiple. Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, ... Multiples of 14: 14, 28, 42, 56, 70, 84,... The least of the common multiples of two or more numbers is the least common multiple (LCM). The LCM of 6 and 14 is 42. Slide 55 / 69 Find the least common multiple of 18 and 24. Slide 56 / 69 31 Find the least common multiple of 10 and 14. Multiples of 18: 18, 36, 54, 72, ... A 2 Multiples of 24: 24, 48, 72, ... B 20 LCM: 72 C 70 D 140 Slide 57 / 69 32 Find the least common multiple of 5 and 30. Slide 58 / 69 33 Find the least common multiple of 9 and 15. A 6 A 3 B 10 B 30 C 30 C 45 D 150 D 135 Slide 59 / 69 34 Find the least common multiple of 3, 6, and 9. Slide 60 / 69 35 Find the least common multiple of 16, 20, and 30. A 3 A 80 B 12 B 100 C 18 C 240 D 36 D 320 Slide 61 / 69 Slide 62 / 69 Another way to find the least common multiple (LCM) is to factor the numbers into primes and then multiply all of the factors, using each common factor only once. Find the least common multiple (LCM) by factoring the number into primes and then multiply all of the factors, using each common factor only once. Example: Find the LCM of 12 and 18. Example: Find the LCM of 16 and 28. 2 12 2 18 2 6 3 9 3 3 1 33 1 12 = 2 x 2 x 3 2 16 18 = 2 x 3 x 3 2 8 LCM: 2 x 3 x 2 x 3 = 36 16 = 2 x 2 x 2 x 2 2 28 2 14 2 4 2 2 7 7 1 28 = 2 x 2 x 7 LCM: 2 x 2 x 2 x 2 x 7 = 112 1 Slide 63 / 69 Slide 64 / 69 Find the least common multiple (LCM) by factoring the numbers into primes and then multiply all of the factors, using each common factor only once. 36 Use prime factorization to find the LCM of 12 and 20. Example: Find the LCM of 10, 12, and 20. 5 5 1 10 = 2 x 5 2 20 2 12 2 10 2 6 3 3 1 2 10 12 = 2 x 2 x 3 5 5 20 = 2 x 2 x 5 1 LCM: 2 x 5 x 2 x 3 x 5 = 300 Slide 65 / 69 37 Use prime factorization to find the LCM of 24 and 60. Slide 66 / 69 38 Use prime factorization to find the LCM of 9, 15, and 18. Slide 67 / 69 39 Use prime factorization to find the LCM of 16, 24, and 32. Slide 69 / 69 41 Use prime factorization to find the GCF of 15, 20, 75. Slide 68 / 69 40 Use prime factorization to find the LCM of 15, 20, 75.