NS2.4 Determine the least common multiple and the greatest common divisor of whole numbers; use them to solve problems with fractions (e.g. to find a common denominator to add two fractions or to find the reduced form of a fraction). California Standards

VOCABULARY CARDS Divisor Greatest Common Divisor (GCD)

The greatest common divisor (GCD) of two or more whole numbers is the greatest whole number that divides evenly into each number. One way to find the GCD of two or more numbers is to list all the factors of each number. The GCD is the greatest factor that appears in all the lists.

Teacher Example 1: Using a List to Find the GCD Find the greatest common divisor (GCD) of 12, 36, and 54. Factors of 12: 1, 2, 3, 4, 6, 12 Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54 List all of the factors of each number. Circle the greatest factor that is in all the lists. The GCD is 6.

Student Practice 1: Find the greatest common divisor of 14, 28, and 63. 14: 1, 2, 7, 14 28: 1, 2, 4, 7, 14, 28 63: 1, 3, 7, 9, 21, 63 List all of the factors of each number. Circle the greatest factor that is in all the lists. The GCD is 7.

Teacher Example 2: Problem Solving Application You have 120 red beads, 100 white beads, and 45 blue beads. You want to use all the beads to make identical bracelets that have red, white, and blue beads on each. What is the greatest number of matching bracelets you can make?

Understand the Problem Teacher Example 2 Continued: 1 Understand the Problem Rewrite the question as a statement. • Find the greatest number of matching bracelets you can make. List the important information: • There are 120 red beads, 100 white beads, and 45 blue beads. • Each bracelet must have the same number of red, white, and blue beads. The answer will be the GCD of 120, 100, and 45.

Teacher Example 2 Continued: Make a Plan You can list the prime factors of 120, 100, and 45 to find the GCD. Solve 3 120 = 2  2  2  3  5 100 = 2  2  5  5 45 = 3  3  5 The GCD of 120, 100, and 45 is 5. You can make 5 bracelets.

Teacher Example 2 Continued: Look Back 4 If you make 5 bracelets, each one will have 24 red beads, 20 white beads, and 9 blue beads, with no beads left over.

Student Practice 2: Nathan has made fishing flies that he plans to give away as gift sets. He has 24 wet flies and 18 dry flies. Using all of the flies, how many sets can he make?

Understand the Problem Student Practice 2 Continued: 1 Understand the Problem Rewrite the question as a statement. • Find the greatest number of sets of flies he can make. List the important information: • There are 24 wet flies and 18 dry flies. • He must use all of the flies. The answer will be the GCD of 24 and 18.

Student Practice 2 Continued: Make a Plan You can list the prime factors of 24 and 18 to find the GCD. Solve 3 24 = 2  2  2  3 18 = 2  3  3 Multiply the prime factors that are common to both 24 and 18. 2  3 = 6 You can make 6 sets of flies.

Student Example 2 Continued: Look Back 4 If you make 6 sets, each set will have 3 dry flies and 4 wet flies.

Lesson Quiz Find the greatest common divisor (GCD). 1. 28, 40 2. 24, 56 3. 54, 99 4. 20, 35, 70 4 8 9 5 5. The math clubs from 3 schools agreed to a competition. Members from each club must be divided into teams, and teams from all clubs must be equally sized. What is the greatest number of members that can be on a team if Georgia has 16 members, Williams has 24 members, and Fulton has 72 members? 8