Learn to find the greatest common factor (GCF) of a set of numbers.

Vocabulary greatest common factor (GCF)‏

Factors shared by two or more whole numbers are called common factors Factors shared by two or more whole numbers are called common factors. The largest of the common factors is called the greatest common factor, or GCF. Factors of 24: Factors of 36: Common factors: 1, 2, 3, 4, 6, 8, 12, 24 1, 2, 3, 4, 6, 9, 12, 18, 36 1, 2, 3, 4, 6, 12 The greatest common factor (GCF) of 24 and 36 is 12. Example 1 shows three different methods for finding the GCF.

Additional Example 1A: Finding the GCF Find the GCF of the set of numbers. 28 and 42 Method 1: List the factors. factors of 28: factors of 42: List all the factors. 1, 2, 4, 7, 14, 28 1, 2, 3, 6, 7, 14, 21, 42 Circle the GCF. The GCF of 28 and 42 is 14.

Additional Example 1B: Finding the GCF Find the GCF of the set of numbers. 18, 30, and 24 Method 2: Use the prime factorization. 18 = 30 = 24 = 2 • 3 • 3 Write the prime factorization of each number. 2 • 3 • 5 2 • 3 • 2 • 2 Find the common prime factors. Find the prime factors common to all the numbers. 2 • 3 = 6 The GCF of 18, 30, and 24 is 6.

Additional Example 1C: Finding the GCF Find the GCF of the set of numbers. 45, 18, and 27 Method 3: Use a ladder diagram. 3 45 18 27 Begin with a factor that divides into each number. Keep dividing until the three have no common factors. 3 15 6 9 5 2 3 Find the product of the numbers you divided by. 3 • 3 = 9 The GCF of 45, 18, and 27 is 9.

Find the GCF of the set of numbers. Check It Out: Example 1A Find the GCF of the set of numbers. 18 and 36 Method 1: List the factors. factors of 18: factors of 36: List all the factors. 1, 2, 3, 6, 9, 18 1, 2, 3, 4, 6, 9, 12, 18, 36 Circle the GCF. The GCF of 18 and 36 is 18.

Find the GCF of the set of numbers. Check It Out: Example 1B Find the GCF of the set of numbers. 10, 20, and 30 Method 2: Use the prime factorization. 10 = 20 = 30 = 2 • 5 Write the prime factorization of each number. 2 • 5 • 2 2 • 5 • 3 Find the common prime factors. Find the prime factors common to all the numbers. 2 • 5 = 10 The GCF of 10, 20, and 30 is 10.

Find the GCF of the set of numbers. Check It Out: Example 1C Find the GCF of the set of numbers. 40, 16, and 24 Method 3: Use a ladder diagram. 2 40 16 24 Begin with a factor that divides into each number. Keep dividing until the three have no common factors. 2 20 8 12 2 10 4 6 5 2 3 Find the product of the numbers you divided by. 2 • 2 • 2 = 8 The GCF of 40, 16, and 24 is 8.

Additional Example 2: Problem Solving Application Jenna has 16 red flowers and 24 yellow flowers. She wants to make bouquets with the same number of each color flower in each bouquet. What is the greatest number of bouquets she can make?

Understand the Problem 1 Understand the Problem The answer will be the greatest number of bouquets 16 red flowers and 24 yellow flowers can form so that each bouquet has the same number of red flowers, and each bouquet has the same number of yellow flowers. 2 Make a Plan You can make an organized list of the possible bouquets.

3 2 Solve 3 Bouquets Yellow Red 4 Look Back RR YYY RR YYY RR YYY RR YYY RR YYY RR YYY RR YYY RR YYY 16 red, 24 yellow: Every flower is in a bouquet The greatest number of bouquets Jenna can make is 8. 4 Look Back To form the largest number of bouquets, find the GCF of 16 and 24. factors of 16: factors of 24: 1, 2, 4, 8, 16 1, 2, 3, 4, 6, 8, 12, 24 The GCF of 16 and 24 is 8.

Check It Out: Example 2 Peter has 18 oranges and 27 pears. He wants to make fruit baskets with the same number of each fruit in each basket. What is the greatest number of fruit baskets he can make?

Understand the Problem Check It Out: Example 2 Continued 1 Understand the Problem The answer will be the greatest number of fruit baskets 18 oranges and 27 pears can form so that each basket has the same number of oranges, and each basket has the same number of pears. 2 Make a Plan You can make an organized list of the possible fruit baskets.

3 2 3 Solve The greatest number of baskets Peter can make is 9. 4 Bouquets Pears Oranges OO PPP OO PPP OO PPP OO PPP OO PPP OO PPP OO PPP OO PPP OO PPP 18 oranges, 27 pears: Every fruit is in a basket The greatest number of baskets Peter can make is 9. 4 Look Back To form the largest number of baskets, find the GCF of 18 and 27. factors of 18: factors of 27: 1, 2, 3, 6, 9, 18 1, 3, 9, 27 The GCF of 18 and 27 is 9.