Slide 1 / 69 Whole Numbers Table of Contents Slide 2 / 69 Prime and Composite Numbers Prime Factorization Common Factors Greatest Common Factor Relatively Prime Least Common Multiple Slide 3 / 69 Prime and Composite Numbers
Slide 4 / 69 1 The smallest prime number is. Slide 5 / 69 2 49 is not a prime number. Slide 6 / 69 True False
3 This list contains 3 prime numbers: 1, 2, 3, 5, 9, and 12 True Slide 7 / 69 False 4 This list contains 3 prime numbers: 5, 9, 20, 31, 42, 53, and 63 True Slide 8 / 69 False 5 This list contains 3 prime numbers: 5, 9, 20, 31, 42, 53, and 63 True Slide 9 / 69 False
6 This list contains 3 prime numbers: 15, 19, 23, 37, 47, 55, and 63 True Slide 10 / 69 False 7 This list contains 3 prime numbers: 25, 29, 33, 38, 45, 57, and 76 True Slide 11 / 69 False The Sieve of Erastosenes Find the prime numbers by sifting out the multiples of each prime. Slide 12 / 69 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?
The Sieve of Erastosenes Slide 13 / 69 Sift out the multiples of each prime. What are you left with? A Composite Number can be divided evenly by numbers other than 1 or itself. Slide 14 / 69 Examples: 1 is NOT composite. Why not? X Is 18 prime or composite? Explain Is 63 prime or composite? Explain Slide 15 / 69 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 16 / 69 8 43 is Slide 17 / 69 A B Prime Composite 9 30 is Slide 18 / 69 A B Prime Composite
10 33 is Slide 19 / 69 A B Prime Composite Slide 20 / 69 Factoring a Number Factors Slide 21 / 69 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?
Slide 22 / 69 Prime Factorization is the process of factoring a number so that all of the factors are prime numbers. Process for factoring a number into primes Slide 23 / 69 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: 2 12 12 = 2 x 2 x 3 = 2 2 x 3 2 3 6 3 1 What is the prime factorization of 18? Slide 24 / 69 2 18 3 9 3 3 1 18 = 2 x 3 x 3 click for = 2 x 3 2 answer
What is the prime factorization of 24? Slide 25 / 69 2 24 2 12 2 6 3 3 1 24 = 2 x 2 x 2 x 3 click = 2 3 x 3 for answer 11 What is the prime factorization of 30? Slide 26 / 69 A 2 x 3 x 5 B 6 x 5 C 5 x 6 D 2 x 15 12 What is the prime factorization of 24? Slide 27 / 69 A 3 x 8 B 2 x 2 x 6 C 2 3 x 3 D 2 x 2 x 2 x 3
13 What is the prime factorization of 45? Slide 28 / 69 A 3 x 15 B 3 2 x 5 C 9 x 5 D 5 2 x 3 14 What is the prime factorization of 60? Slide 29 / 69 A 2 x 3 x 10 B 2 x 5 x 2 x 3 C 2 2 x 3 x 5 D 2 2 x 15 15 What is the prime factorization of 100? Slide 30 / 69 A 2 x 3 x 10 B 2 x 5 x 2 x 3 C 2 2 x 3 x 5 D 2 2 x 15
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. Slide 31 / 69 Factors of 12: 1, 2, 3, click 4, 6, for 12answer Factors of 16: 1, 2, 4, click 8, 16 for answer Common factors: 1, 2, 4 click for answer What is the Greatest Common Factor? Greatest Common Factor: 4 click for answer Common Factors Find the common factors of 18 and 24. Slide 32 / 69 Factors of 18: 1, 2, 3, click 6, 9, for 18answer Factors of 24: 1, 2, 3, click 4, 6, for 8,12, answer 24 Common factors: 1, 2, 3, 4, 6 click for answer What is the Greatest Common Factor? Greatest Common Factor: 6 click for answer 16 The greatest common factor for 12 and 48 is. Slide 33 / 69 A 2 B 4 C 6 D 12
17 The greatest common factor for 24 and 36 is. Slide 34 / 69 A 2 B 4 C 6 D 12 18 The greatest common factor for 42 and 64 is. Slide 35 / 69 A 2 B 4 C 6 D 8 19 The greatest common factor for 50 and 100 is. Slide 36 / 69 A 5 B 10 C 25 D 50
20 The greatest common factor for 36 and 90 is. Slide 37 / 69 A 3 B 9 C 12 D 18 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. Greatest Common Factor Slide 38 / 69 3. Multiply the common factors together to find the greatest common factor. Slide 39 / 69
Slide 40 / 69 Slide 41 / 69 21 Use prime factorization to find the GCF of 18 and 44. Slide 42 / 69
22 Use prime factorization to find the GCF of 28 and 70. Slide 43 / 69 23 Use prime factorization to find the GCF of 55 and 110. Slide 44 / 69 24 Use prime factorization to find the GCF of 52 and 78. Slide 45 / 69
25 Use prime factorization to find the GCF of 72 and 75. Slide 46 / 69 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. 26 Identify at least two numbers that are relatively prime to 9 Slide 48 / 69 A 16 B 15 C 28 D 36
27 7 and 35 are not relatively prime. Slide 49 / 69 True False 28 Name a number that is relatively prime to 20. Slide 50 / 69 29 Name a number that is relatively prime to 5 and 18. Slide 51 / 69
30 Find two numbers that are relatively prime Slide 52 / 69 A 7 B 14 C 15 D 49 Slide 53 / 69 Least Common Multiple A multiple of a whole number is the product of the number and any nonzero whole number. Slide 54 / 69 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.
Find the least common multiple of 18 and 24. Slide 55 / 69 Multiples of 18: 18, 36, 54, 72,... Multiples of 24: 24, 48, 72,... LCM: 72 31 Find the least common multiple of 10 and 14. Slide 56 / 69 A 2 B 20 C 70 D 140 32 Find the least common multiple of 5 and 30. Slide 57 / 69 A 6 B 10 C 30 D 150
33 Find the least common multiple of 9 and 15. Slide 58 / 69 A 3 B 30 C 45 D 135 34 Find the least common multiple of 3, 6, and 9. Slide 59 / 69 A 3 B 12 C 18 D 36 35 Find the least common multiple of 16, 20, and 30. Slide 60 / 69 A 80 B 100 C 240 D 320
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. Slide 61 / 69 Example: Find the LCM of 12 and 18. 2 12 2 18 12 = 2 x 2 x 3 2 6 3 9 18 = 2 x 3 x 3 3 3 1 3 3 1 LCM: 2 x 3 x 2 x 3 = 36 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. Slide 62 / 69 Example: Find the LCM of 16 and 28. 2 16 2 28 16 = 2 x 2 x 2 x 2 2 8 2 14 28 = 2 x 2 x 7 2 4 2 2 7 7 1 LCM: 2 x 2 x 2 x 2 x 7 = 112 1 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. Slide 63 / 69 Example: Find the LCM of 10, 12, and 20. 2 10 5 5 1 2 12 2 6 3 3 2 20 2 10 5 5 10 = 2 x 5 12 = 2 x 2 x 3 20 = 2 x 2 x 5 1 1 LCM: 2 x 5 x 2 x 3 x 5 = 300
36 Use prime factorization to find the LCM of 12 and 20. Slide 64 / 69 37 Use prime factorization to find the LCM of 24 and 60. Slide 65 / 69 38 Use prime factorization to find the LCM of 9, 15, and 18. Slide 66 / 69
39 Use prime factorization to find the LCM of 16, 24, and 32. Slide 67 / 69 40 Use prime factorization to find the LCM of 15, 20, 75. Slide 68 / 69 41 Use prime factorization to find the GCF of 15, 20, 75. Slide 69 / 69