The Number System Greatest Common Factor 1 © 2013 Meredith S. Moody

Objective: You will be able to… Find all the factors of a number Prime factor a number Find the greatest common factor for a pair or set of numbers 2 © 2013 Meredith S. Moody

Vocabulary Factor: Any number that is divisible by itself (or another whole number larger than itself) a whole number of times Any two numbers are factors of another number when they are multiplied together and their product is that number 3 © 2013 Meredith S. Moody

Example 1, factoring The number 6 has four factors: 1, 2, 3, & 6 1 x 6 = 6 2 x 3 = 6 1, 2, 3, & 6 are factors of 6 4 © 2013 Meredith S. Moody

Guided practice 1, finding factors Find the factors of the number 10 1 x ? = 10  1 x 10 = 10 2 x ? = 10  2 x 5 = 10 3 x ? = 10  none 4 x ? = 10  none 5 x ? = 10  5 x 2 = 10  we already have 5 & 2, so we have found all the factors of 10  1, 2, 5, and 10 5 © 2013 Meredith S. Moody

Vocabulary Greatest common factor : The number with the largest value that is a factor for more than one number (a set or pair) 6 © 2013 Meredith S. Moody

Example 1, common factors, ‘list’ method The numbers 6 and 8 have a common factor The factors of 6 are 1, 2, 3, & 6 The factors of 8 are 1, 2, 4, & 8 The common factor of 6 and 8 is 2 We do not include ‘1’ because it is a factor of all other numbers 7 © 2013 Meredith S. Moody

Example 1, greatest common factor, ‘list’ method The numbers 28 and 44 have more than one common factor The factors of 28 are 1, 2, 4, 7, 14, and 28 The factors of 44 are 1, 2, 4, 11, and 22 The common factors of 28 and 44 are 2 & 4 The greatest common factor of 28 & 44 is 4 8 © 2013 Meredith S. Moody

Guided practice 1, greatest common factor, ‘list’ method Find the greatest common factor of the numbers 30 and 50 What are the factors of 30? 1, 2, 3, 5, 6, 10, 15 What are the factors of 50? 1, 2, 5, 10, 25, 50 What are the numbers that are factors of both numbers? 2, 5, & 10 What is the greatest common factor? 10 9 © 2013 Meredith S. Moody

Vocabulary Prime number: A number with ONLY 2 factors: 1 and the number itself. For example, 5  only 1 and 5 are the factors of 5  5 is a prime number The number “1” is a special case 10 © 2013 Meredith S. Moody

Example, prime numbers The number 7 is a prime number Only 1 x 7 = 7 There are no other numbers whose product is 7 Therefore, 7 is a prime number 11 © 2013 Meredith S. Moody

Prime numbers: You try Give 3 examples of prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29… How do you know they are prime? The only factors are 1 and itself 12 © 2013 Meredith S. Moody

Vocabulary Prime factorization : Breaking a number down into factors that are all prime numbers 13 © 2013 Meredith S. Moody

Example 1, prime factorization Two factors of 12 are 3 and 4 3 x 4 = 12 3 is a prime number Is 4 a prime number? No 2 x 2 = 4; is 2 a prime number? Yes Therefore, 3 x 2 x 2 = 12 3 and 2 are prime numbers, so the prime factorization of 12 is 3 x 2 x 2 14 © 2013 Meredith S. Moody

Example 1, continued Two other factors of 12 are 2 and 6 2 x 6 = 12 2 is a prime number Is 6 a prime number? No 2 x 3 = 6 2 is prime; is 3 a prime number? Yes Therefore, 2 x 2 x 3 = 12 2 and 3 are prime numbers, so the prime factorization of 12 is 2 x 2 x 3 no matter which factors you begin with 15 © 2013 Meredith S. Moody

Vocabulary Factor tree: A diagram used to break down a number by dividing it by its factors until all the numbers left are prime © 2013 Meredith S. Moody16

Example 1, factor trees Factor trees are a useful tool when prime factoring a number Here is a factor tree for the number 42 The prime factorization is 2 x 3 x 7 17 © 2013 Meredith S. Moody

Example 2, factor trees Here is a longer factor tree Notice how the prime numbers are circled as they are found The prime factorization of 72 is 2 x 2 x 2 x 3 x 3 18 © 2013 Meredith S. Moody

Factor trees: You try Make a factor tree for the number 108 and list the prime factorization The prime factorization is 2 x 2 x 3 x 3 x 3 19 © 2013 Meredith S. Moody

Guided practice 1, prime factorization Find the prime factorization of 50 5 x 10 = 50 Is 5 a prime number? Yes Is 10 a prime number? No 2 x 5 = 10 Is 2 a prime number? Yes; is 5 a prime number? Yes What is the prime factorization of 50? 2 x 5 x 5 = © 2013 Meredith S. Moody

Prime factorization: You try Find the prime factorization of 32 The prime factorization of 32 is 2 x 2 x 2 x 2 x 2 21 © 2013 Meredith S. Moody

Guided practice 1, greatest common factor, ‘prime product’ method You can use the prime factorization to find the greatest common factor Find the prime factorization of both numbers and multiply the common prime factors Find the greatest common factor of 63 and 84 using the prime factorization method 22 © 2013 Meredith S. Moody

Guided practice 1, greatest common factor, ‘prime product’ method, continued What is the prime factorization of 63? What is the prime factorization of 84? © 2013 Meredith S. Moody 23

Guided practice 1, greatest common factor, ‘prime product’ method, continued What are the common prime factors of 63 and 84? 3 and 7 What is the product of those common prime factors? 3 x 7 = 21 What is the greatest common factor of 63 and 84? © 2013 Meredith S. Moody

Which is better? Why would you tell someone to use the list method if they want to find the greatest common factor? Why would you tell someone to use the prime product method if they want to find the greatest common factor? Is there a time one is better than the other? The list method works better with smaller numbers The prime product works better with larger numbers Does it matter which one you use? Technically, no – although using the appropriate method will save you time 25 © 2013 Meredith S. Moody

Greatest common factors of number sets Use the same process to find common factors even if you are working with more than two numbers 26 © 2013 Meredith S. Moody

Example 1, greatest common factor of a number set, ‘list’ method Find the greatest common factor for the number set {15, 75, 150} Factors of 15 are 1, 3, 5, & 15 Factors of 75 are 1, 3, 5, 15, 25, & 75 Factors of 150 are 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, & 150 The common factors are 3, 5, & 15 The greatest common factor is © 2013 Meredith S. Moody

Example 2, greatest common factor of a number set, ‘prime product’ method Find the greatest common factor of the number set {84, 108, 216} First, prime factor 84 © 2013 Meredith S. Moody 28

Example 2, greatest common factor of a number set, ‘prime product’ method, continued Next, prime factor © 2013 Meredith S. Moody

Example 2, greatest common factor of a number set, ‘prime product’ method, continued Then, prime factor © 2013 Meredith S. Moody

Example 2, greatest common factor of a number set, ‘prime product’ method, continued 2 x 2 x 3 x 7 = 84 2 x 2 x 3 x 3 x 3 = x 2 x 2 x 3 x 3 x 3 = 216 All three numbers have 2 x 2 x 3 in common; 2 x 2 x 3 = 12 The greatest common factor of the number set {84, 108, 216} is © 2013 Meredith S. Moody

Guided practice 1, greatest common factor Find the greatest common factor using the list method: 24 and 40 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Factors of 40: 1, 2, 4, 5, 9, 10, 20, 40 Common factors: 1, 2, 4 Greatest common factor: 4 32 © 2013 Meredith S. Moody

Guided practice 2, greatest common factor Find the greatest common factor using the prime product method: 165 and 135 Common factors: 3, 5 3 x 5 = 15 Greatest common factor: 15 © 2013 Meredith S. Moody33

Greatest common factor: You try #1 Find the greatest common factor using any method: 18 and and and © 2013 Meredith S. Moody

Guided practice 3, greatest common factor Bennie is catering a party. He has 60 celery sticks and 45 small tacos. He wants both kinds of food in each plate. He wants the food distributed evenly and none left over: what is the largest number of plates he can use and how many of each type of food will be on each plate? 35 © 2013 Meredith S. Moody

Guided practice 3, greatest common factor, continued The greatest number of plates is going to be the greatest common factor of 60 and 45, because that way, there will be no food left off any of the plates Find the greatest common factor of 60 and © 2013 Meredith S. Moody

Guided practice 3, greatest common factor, continued Remember, Bennie wants the food evenly distributed If there are 15 plates, how many celery sticks will go on each plate? 60 ÷ 15 = 4; 4 celery sticks on each plate If there are 15 plates, how many small tacos will go on each plate? 45 ÷ 15 = 3; 3 tacos on each plate 37 © 2013 Meredith S. Moody

Greatest common factor, You try #2 Jane is putting together candy bags for her Halloween party The candy bags are sold in bundles of 15, 20, and 25 She buys 3 bags of mini Reese’s candy with 40 cups per bag She buys 2 bags of mini Hershey’s candy with 25 bars per bag She buys 2 bags of mini Snicker’s candy with 30 bars per bag What is the greatest number of bags she should buy and how much of each type of candy should she put in each bag if she wants an even distribution? Will there be any bags left over? If so, how many? Will there be any candy left over? If so, what kind and how many? 38 © 2013 Meredith S. Moody

Answer Jane has 120 Reese’s, 50 Hershey’s, and 60 Snickers The greatest common factor of 120, 50, and 60 is 10 She should buy the 15 bag pack and put 12 Reese’s, 5 Hershey’s, and 6 Snickers in each bag There will be 5 bags left over There will be no candy left over 39 © 2013 Meredith S. Moody