Core Focus on Decimals & Fractions Lesson 3.1 Core Focus on Decimals & Fractions Greatest Common Factor

Warm-Up 1. Write three multiplication problems using whole numbers that equal 20. 2. Find the value of 3 × 5. 3. Find the value of 6 × 7. 1  20 2  10 4  5 15 42

Greatest Common Factor Lesson 3.1 Greatest Common Factor Find the greatest common factor (GCF) of a set of numbers.

Vocabulary Factors Numbers that can be multiplied to find a product. Prime Number A whole number that has only two possible factors (1 and itself). Composite Number A whole number larger than one with more than two factors.

These are the same as the first column but in reverse order Example 1 Determine if 12 is a prime or composite number. List the pairs of numbers that 1 × 12 12 × 1 have a product of 12. 2 × 6 6 × 2 3 × 4 4 × 3 List each factor once, even 1, 2, 3, 4, 6, 12 if it is repeated. The factors of 12 are 1, 2, 3, 4, 6, and 12. There are more than two factors so the number 12 is composite. These are the same as the first column but in reverse order

Vocabulary Continued… Greatest Common Factor The greatest factor that is a whole number common to all the numbers.

Explore! University Sales Bracken had 36 University of Miami shirts and 42 Florida State University shirts to sell. He wants to stack them in piles that would all have the same number of shirts. He does not want to mix the two types of shirts. What is the greatest number of shirts that can be stacked in each pile? Find the GCF of 36 and 42. Step 1 Find all factors of 36 by filling the boxes with the missing factors. Make a list of all factors of 36.   36 2     12 4     6 Step 2 Find all factors of 42 by filling the boxes with the missing factors. Make a list of all factors of 42. 1   2   3     7 Step 3 Circle the common factors. Common factors are factors that are the same for both 36 and 42.

Explore! University Sales Step 4 Draw a Venn diagram like the one below on a sheet of paper. Write “Factors of 36” on the outside of the left circle and “Factors of 42” on the outside of the right circle. Factors of 36 Factors of 42 Step 5 Write all factors in the Venn diagram. Write the factors that both numbers have in common in the overlapping part of the circles. The remaining factors of 36 should be written in the yellow part of the left circle. The remaining factors of 42 should be written in the pink part of the right circle.

Explore! University Sales Step 6 Look at the common factors written in the overlapping part of the circles in the Venn diagram. Circle the largest number. This is the greatest common factor (GCF). Step 7 Use the GCF to answer the question in the problem at the beginning of the Explore! in a complete sentence. Step 8 Repeat Steps 1–6 to find the GCF of the following pairs of numbers: a. 15 and 25 b. 18 and 30 c. 24 and 40

Vocabulary Continued… Prime Factorization A composite number written as a product of all its prime factors.

Example 2 Team 1 36 Players Team 2 30 Players Two local teams went to soccer camp together. At the camp the teams were asked to split into equal amounts for cabin groups. The players did not want to room with players from other teams. The camp directors want the largest number possible in each cabin. How many players will be in each cabin? Use prime factors to find the GCF. Prime factors are factors that are prime numbers. Write each number as products of two factors. Continue to write each number as products of two factors until only factors that are prime numbers remain. Team 1 36 Players Team 2 30 Players 36 4  9 2  2 3  3 30 6  5 2  3

Example 2 Continued… Team 1 36 Players Team 2 30 Players Two local teams went to soccer camp together. At the camp the teams were asked to split into equal amounts for cabin groups. The players did not want to room with players from other teams. The camp directors want the largest number possible in each cabin. How many players will be in each cabin? Write the factors out for each number. 36 = 2  2  3  3 30 = 2  3  5 This is called the prime factorization. Highlight the common prime factors. Find the product of the common prime GCF = 2  3 = 6. The GCF is 6. factors. This is the GCF. Six players will be in each cabin. 36 4  9 2  2 3  3 30 6  5 2  3 Team 1 36 Players Team 2 30 Players

Example 3 Reagan Middle School students were asked to sit in equal rows for the assembly. There were 98 sixth graders, 84 seventh graders and 112 eighth graders. The teacher did not want grade levels to sit together, but the rows were to be as wide as possible. How many students should sit in each row? List the factors of each number. Highlight the common factors. Factors of 98: 1, 2, 7, 14, 49, 98 Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 Factors of 112: 1, 2, 4, 7, 8, 14, 16, 28, 56, 112 Find the GCF. Fourteen students should sit in each row.

Finding the Greatest Common Factor LIST PRIME FACTORIZATION 1. List the factors for each number. 2. Highlight the common factors. 3. Identify the GCF (greatest common factor). 1. Write each number as a product of its prime factors. 2. Highlight the common prime factors. 3. Find the product of the common prime factors to identify the GCF (greatest common factor).

Communication Prompt What is a real life situation in which you would need to find the greatest common factor?

Exit Problems 1. Find the GCF of 56 and 64. 2. Gina wants to sell 49 chocolate chip cookies and 35 sugar cookies. She is going to sell them on plates with equal amounts on each plate. Each plate needs to hold the largest number of cookies without mixing types of cookies. How many cookies should Gina put on each plate? 8 7 cookies