Greatest Common Factor (GCF)

Show your work for each problem.

1.
What is the GCF of $6$ and $9$ ?
1. a)9
2. b)6
3. c)18
4. d)3
Answer: 3
Factors of 6: 1, 2, 3, 6. Factors of 9: 1, 3, 9. Common factors: 1, 3. The greatest is $3$ .
2.
What is the GCF of $8$ and $12$ ?
Answer: 4
Factors of 8: 1, 2, 4, 8. Factors of 12: 1, 2, 3, 4, 6, 12. Common factors: 1, 2, 4. GCF = $4$ .
3.
Find the GCF of $15$ and $20$ 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)6 and 9
2. b)8 and 15
3. c)10 and 25
4. d)12 and 18
Answer: 8 and 15
GCF(8, 15) = 1 (no common factor other than 1). 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 $24$ and $36$ ?
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 = $12$ .
6.
Find the GCF of $48$ and $60$ using prime factorization.
Answer: 12
1. Write it with exponents: $48 = 2^{\square} \times 3$ . What exponent goes in the box? 4
2. Write it with exponents: $60 = 2^{\square} \times 3 \times 5$ . What exponent goes in the box? 2
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 $\frac{18}{24}$ using the GCF.
1. a) $\frac{9}{12}$
2. b) $\frac{6}{8}$
3. c) $\frac{3}{4}$
4. d) $\frac{2}{3}$
Answer: $\frac{3}{4}$
GCF(18, 24) = 6. $\frac{18 \div 6}{24 \div 6} = \frac{3}{4}$ . 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. Do you need the GCF or the LCM here? 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 GCF(36, 48), the maximum number of bouquets? 12
9.
What is the GCF of $42$ and $56$ ?
Answer: 14
$42 = 2 \times 3 \times 7$ , $56 = 2^3 \times 7$ . Common: $2^1$ and $7^1$ . GCF = $2 \times 7 = 14$ .
10.
What is GCF(50, 100)?
1. a)50
2. b)10
3. c)25
4. d)100
Answer: 50
Since 50 divides evenly into 100 ( $100 = 2 \times 50$ ), the GCF is 50 itself. The GCF can equal the smaller number!
11.
Find the GCF of $18$ , $24$ , and $30$ .
Answer: 6
1. Write it with exponents: $18 = 2 \times 3^{\square}$ . What exponent goes in the box? 2
2. Write it with exponents: $24 = 2^{\square} \times 3$ . What exponent goes in the box? 3
3. Factor 30 into primes: $30 = 2 \times 3 \times \square$ . Which prime goes in the box? 5
4. Which primes appear in ALL three factorizations? 2 and 3
5. GCF = ? 6
12.
What is GCF(72, 96)?
Answer: 24
$72 = 2^3 \times 3^2$ , $96 = 2^5 \times 3$ . Common: $2^3$ and $3^1$ . GCF = $8 \times 3 = 24$ .
13.
Simplify $\frac{84}{126}$ to lowest terms using the GCF.
Answer: 2/3
1. Write it with exponents: $84 = 2^{\square} \times 3 \times 7$ . What exponent goes in the box? 2
2. Write it with exponents: $126 = 2 \times 3^{\square} \times 7$ . What exponent goes in the box? 2
3. GCF(84, 126) = ? 42
4. $\frac{84}{126} = \frac{\square}{3}$ . What number goes in the box? 2
14.
A carpenter has boards of 84 cm and 126 cm. She wants to cut both into pieces of the same length with no waste. What is the longest possible piece?
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 the 84 cm board and 3 from the 126 cm board, all 42 cm long.
15.
Find the GCF of $10$ and $15$ .
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
$45 = 3^2 \times 5$ , $75 = 3 \times 5^2$ . Common: $3^1$ and $5^1$ . GCF = $3 \times 5 = 15$ .