6.1.5 Greatest Common Factor

Vocabulary Common Factors – Factors shared by two or more numbers Example: 5 is a common factor of 30 and 25 7 is a common factor of 21 and 42 Greatest Common Factor – The largest factor shared by two numbers Example: It is true that 2 is a common factor of 16 and 24, but their GREATEST common factor is actually 8 9 is the greatest common factor of 18 and 36 12 is the greatest common factor of 24 and 48 GCF = Greatest Common Factor

How to Find the Greatest Common Factor? Venn Diagram Prime Factorization

Venn Diagram Example Book Something you read Movie you watch Tells a story Is entertaining

Greatest Common Factor Venn Diagram

Greatest Common Factor = 12 Greatest Common Factor = 15 Tables Work, Too! Factors of ONLY 36 Factors of 36 AND 48 Factors of ONLY 48 9 18 36 12 4 3 1 Greatest Common Factor = 12 6 24 16 48 Factors of ONLY 30 Factors of 30 AND 75 Factors of ONLY 75 10 2 30 15 5 3 Greatest Common Factor = 15 75

Your Turn! Factors of ONLY 16 Factors of 16 AND 56 Factors of ONLY 56

Greatest Common Factor = 8 Your Turn! Factors of ONLY 16 Factors of 16 AND 56 Factors of ONLY 56 16 8 4 2 Greatest Common Factor = 8 14 56

Compare Three Numbers Factors of ONLY 14 Factors of ONLY 28 2 14 49 Common Factors of 14, 28, AND 49 1 and 7

Using Prime Factorization to Find GCF Step #1: Draw out the prime factorization for each number Step #2: Write out the prime factors each number has in common Step #3: Multiply those common prime factors Step #4: The product is the greatest common factor for the two numbers

Find the GCF Using Prime Factorization 12 and 56

Find the GCF Using Prime Factorization 12 = 4 x 3 Prime Factorization = 3 x 2 x 2 4 = 2 x 2 56 = 7 x 8 Prime Factorization = 7 x 2 x 2 x 2 8 = 2 x 4 Common Factors = 2, 2 Greatest Common Factor = 2 x 2 = 4

Using a Venn Diagram of Prime Factors What two numbers does this Venn Diagram represent? 2 3 Step #1: Multiply the numbers in the first circle with the numbers in the overlapping circles (example: 2 x 3 x 3 = 18) Step #2: Multiply the numbers in the second circle with the numbers in the overlapping cirles (example: 3 x 3 x 3 = 27) 3 3 The two numbers are 18 and 27

Your Turn! 2 2 3 3 11 5

The numbers are 180 and 55 2 2 11 3 3 5 2 x 2 x 3 x 3 x 5 = 180 2 2 3 3 11 5 2 x 2 x 3 x 3 x 5 = 180 5 x 11 = 55

Common Prime Factors 3 2 3 3 2 2 3 2 2 2 2 36 = 12 x 3 48 = 12 x 4 Prime Factors of ONLY 36 Common Prime Factors of 36 AND 48 Prime Factors of ONLY 48 3 3 2 2 2 2 36 = 12 x 3 12 = 4 x 3 4 = 2 x 2 36 = 3 x 3 x 2 x 2 48 = 12 x 4 12 = 4 x 3 4 = 2 x 2 48 = 3 x 2 x 2 x 2 x 2 3 2 3 3 2 2 3 2 2 2 2

How Many Gift Bags Can I Make I went to the store to buy small gifts for the gift bags at my son’s birthday party. Unfortunately, I wasn’t able to find the same amount of each item: 18 jars of moon sand (all different colors), 24 Lego packs (all different sets), and a variety pack of Hershey’s mini chocolate bars (42 candy bars in the bag). I want identical groups of gifts in each bag with no gifts left over. What is the greatest number of gift bags I can make? 18 = 2 x 3 x 3 24 = 2 x 3 x 2 x 2 42 = 2 x 3 x 7 2 x 3 = 6 Gift Bags Note: To find out how many of each toy should go in each bag, divide the total number of each toy by the number of gift bags you’re making. For example, of the 42 candy mini candy bars, 7 of them should be placed in each of the 6 gift bags.

Your Turn – Rose Bouquets A florist is making identical bouquets using 72 red roses, 60 pink roses, and 48 yellow roses. What is the greatest number of bouquets that the florist can make if no roses are left over? How many of each color are in each bouquet?

Your Turn – Rose Bouquets 72 = 8 x 9 60 = 10 x 6 48 = 6 x 8 8 = 4 x 2 10 = 5 x 2 6 = 3 x 2 4 = 2 x 2 6 = 3 x 2 8 = 4 x 2 9 = 3 x 3 (5 x 2 x 3 x 2) 4 = 2 x 2 (2 x 2 x 2 x 3 x 3) (3 x 2 x 2 x 2 x 2) Common Prime Factors = 3, 2, and 2 Multiply the common prime factors: 3 x 2 x 2 = 12 Rose Bouquets 72 ÷ 12 = 6 Red Roses 60 ÷ 12 = 5 Pink Roses 48 ÷ 12 = 4 Yellow Roses

Homework Page 34 (# 5, 10, 12, 15, 16, 19, 21, 24, 25, 27)