Greatest Common Factor (GCF)

Show your work for each problem.

1.
What is the GCF of and ?
1. a)6
2. b)3
3. c)18
4. d)9
Answer: 3
Factors of 6: 1, 2, 3, 6. Factors of 9: 1, 3, 9. Common factors: 1, 3. The greatest is .
2.
What is the GCF of and ?
Answer: 4
Factors of 8: 1, 2, 4, 8. Factors of 12: 1, 2, 3, 4, 6, 12. Common factors: 1, 2, 4. GCF = .
3.
Find the GCF of and by listing factors.
Answer: 5
1. List all factors of 15: 1, 3, 5, 15
2. List all factors of 20: 1, 2, 4, 5, 10, 20
3. What factors appear in BOTH lists? 1, 5
4. What is the greatest common factor? 5
4.
Which pair of numbers is relatively prime (GCF = 1)?
1. a)8 and 15
2. b)6 and 9
3. c)10 and 25
4. d)12 and 18
Answer: 8 and 15
GCF(8, 15) = 1 (no common factors). GCF(6, 9) = 3. GCF(10, 25) = 5. GCF(12, 18) = 6. Only 8 and 15 are relatively prime.
5.
What is the GCF of and ?
Answer: 12
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. Common: 1, 2, 3, 4, 6, 12. GCF = .
6.
Find the GCF of and using prime factorization.
Answer: 12
1. Write the prime factorization of 48: 2^4 x 3
2. Write the prime factorization of 60: 2^2 x 3 x 5
3. Which prime factors are common to both? 2 and 3
4. Take the lowest power of each common prime and multiply: 12
7.
Simplify using the GCF.
1. a)
2. b)
3. c)
4. d)
Answer:
GCF(18, 24) = 6. . The other options are not fully simplified.
8.
A florist has 36 roses and 48 tulips. She wants to make identical bouquets with no flowers left over. What is the maximum number of bouquets she can make?
Answer: 12
1. What mathematical operation will help solve this? GCF
2. List factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
3. List factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
4. What is the GCF and maximum number of bouquets? 12
9.
What is the GCF of and ?
Answer: 14
, . Common: and . GCF = .
10.
What is GCF(50, 100)?
1. a)100
2. b)25
3. c)50
4. d)10
Answer: 50
Since 50 divides evenly into 100 (100 = 2 x 50), the GCF is 50 itself. The GCF can equal the smaller number!
11.
Find the GCF of , , and .
Answer: 6
1. Prime factorization of 18: 2 x 3^2
2. Prime factorization of 24: 2^3 x 3
3. Prime factorization of 30: 2 x 3 x 5
4. Common primes with lowest powers: 2^1 x 3^1
5. GCF = ? 6
12.
What is GCF(72, 96)?
Answer: 24
, . Common: and . GCF = .
13.
Simplify to lowest terms using the GCF.
Answer: 2/3
1. Prime factorization of 84: 2^2 x 3 x 7
2. Prime factorization of 126: 2 x 3^2 x 7
3. GCF(84, 126) = ? 42
4. Simplified fraction: 2/3
14.
A carpenter has boards of 84 cm and 126 cm. What is the longest equal piece she can cut from both with no waste?
1. a)21 cm
2. b)7 cm
3. c)42 cm
4. d)14 cm
Answer: 42 cm
GCF(84, 126) = 42. She can cut 2 pieces from 84 cm and 3 pieces from 126 cm, all 42 cm long.
15.
Find the GCF of and .
Answer: 5
1. List factors of 10: 1, 2, 5, 10
2. List factors of 15: 1, 3, 5, 15
3. Common factors: 1, 5
4. Greatest common factor: 5
16.
What is GCF(45, 75)?
Answer: 15
, . Common: and . GCF = .