Using the Greatest Common Divisor (GCD) in Real-World Grouping Problems

1) Two marching bands have 48 and 60 students. They want to line up in rows with the same number of students in each row and no one left out. What is the greatest number of students per row?

6

8

12

15

Factors of 48: {1,2,3,4,6,8,12,16,24,48}. Factors of 60: {1,2,3,4,5,6,10,12,15,20,30,60}. GCD=12.

Explanation

Factors of 48: {1,2,3,4,6,8,12,16,24,48}. Factors of 60: {1,2,3,4,5,6,10,12,15,20,30,60}. GCD=12.

2) A park has 42 benches and 56 trash bins. The city wants to group them into identical sections with no leftovers. What is the largest number of sections?

7

14

21

28

Factors of 42: {1,2,3,6,7,14,21,42}. Factors of 56: {1,2,4,7,8,14,28,56}. GCD=14.

Explanation

Factors of 42: {1,2,3,6,7,14,21,42}. Factors of 56: {1,2,4,7,8,14,28,56}. GCD=14.

3) A farmer has 30 sheep and 45 goats. He wants to arrange them into equal pens, mixing sheep and goats, with no animals left out. What is the greatest number of pens?

5

10

15

20

Factors of 30: {1,2,3,5,6,10,15,30}. Factors of 45: {1,3,5,9,15,45}. GCD=15.

Explanation

Factors of 30: {1,2,3,5,6,10,15,30}. Factors of 45: {1,3,5,9,15,45}. GCD=15.

4) Two playground slides are 48 feet and 64 feet long. The manager wants to build equal-length safety mats for both slides. What is the longest mat length that works for both?

4 ft

8 ft

16 ft

24 ft

GCD of 48 and 64 = 16.

Explanation

GCD of 48 and 64 = 16.

5) A gardener plants 72 roses and 108 tulips in equal rows. What is the maximum number of rows she can make with no leftovers?

12

18

36

72

GCD of 72 and 108 = 18.

Explanation

GCD of 72 and 108 = 18.

6) A coach has 32 soccer balls and 40 cones. She wants to divide them into identical training kits. What is the greatest number of kits she can make?

4

8

10

16

GCD of 32 and 40 = 8.

Explanation

GCD of 32 and 40 = 8.

7) Two buses arrive at a station every 60 minutes and 84 minutes. After how many minutes will both buses be at the station together?

140

420

480

504

LCM(60,84)=420 minutes.

Explanation

LCM(60,84)=420 minutes.

8) A mural wall is 72 cm by 108 cm. The artist wants to tile it with the largest square tiles possible, with no cutting. What is the side length of each square tile?

6 cm

12 cm

36 cm

72 cm

GCD of 72 and 108 = 36 cm.

Explanation

GCD of 72 and 108 = 36 cm.

9) A music teacher has 45 violins and 60 flutes. She wants to form identical orchestra groups. What is the maximum number of groups?

10

12

15

20

GCD of 45 and 60 = 15.

Explanation

GCD of 45 and 60 = 15.

10) A video game tournament has 84 players and 126 players in two divisions. The organizer wants to split them into groups of the same size, with no players left out. What is the largest group size?

14

21

42

63

The largest group size is determined by finding the greatest common divisor of 84 and 126, which is 42.

Explanation

The largest group size is determined by finding the greatest common divisor of 84 and 126, which is 42.

Applying GCD in Real-World Contexts

1) Two marching bands have 48 and 60 students. They want to line up in rows with the same number of students in each row and no one left out. What is the greatest number of students per row?

2) A park has 42 benches and 56 trash bins. The city wants to group them into identical sections with no leftovers. What is the largest number of sections?

3) A farmer has 30 sheep and 45 goats. He wants to arrange them into equal pens, mixing sheep and goats, with no animals left out. What is the greatest number of pens?

4) Two playground slides are 48 feet and 64 feet long. The manager wants to build equal-length safety mats for both slides. What is the longest mat length that works for both?

5) A gardener plants 72 roses and 108 tulips in equal rows. What is the maximum number of rows she can make with no leftovers?

6) A coach has 32 soccer balls and 40 cones. She wants to divide them into identical training kits. What is the greatest number of kits she can make?

7) Two buses arrive at a station every 60 minutes and 84 minutes. After how many minutes will both buses be at the station together?

8) A mural wall is 72 cm by 108 cm. The artist wants to tile it with the largest square tiles possible, with no cutting. What is the side length of each square tile?

9) A music teacher has 45 violins and 60 flutes. She wants to form identical orchestra groups. What is the maximum number of groups?

10) A video game tournament has 84 players and 126 players in two divisions. The organizer wants to split them into groups of the same size, with no players left out. What is the largest group size?