Greatest Common Factor (GCF) A. Find the greatest common factor of the two numbers: 1. 7 and 14 2. 12 and 32 3. 9 and 21 4. 10 and 45 5. 4 and 28
The Correct Answer and Explanation is:
The Greatest Common Factor (GCF) of two numbers is the largest number that divides both numbers without leaving a remainder. To find the GCF, we can list the factors of each number and identify the largest common factor.
1. GCF of 7 and 14:
- Factors of 7: 1, 7
- Factors of 14: 1, 2, 7, 14
- The common factors are 1 and 7. The largest common factor is 7.
2. GCF of 12 and 32:
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 32: 1, 2, 4, 8, 16, 32
- The common factors are 1, 2, and 4. The largest common factor is 4.
3. GCF of 9 and 21:
- Factors of 9: 1, 3, 9
- Factors of 21: 1, 3, 7, 21
- The common factors are 1 and 3. The largest common factor is 3.
4. GCF of 10 and 45:
- Factors of 10: 1, 2, 5, 10
- Factors of 45: 1, 3, 5, 9, 15, 45
- The common factors are 1 and 5. The largest common factor is 5.
5. GCF of 4 and 28:
- Factors of 4: 1, 2, 4
- Factors of 28: 1, 2, 4, 7, 14, 28
- The common factors are 1, 2, and 4. The largest common factor is 4.
Explanation:
To calculate the GCF, we need to identify the common factors of both numbers and choose the largest one. For each pair, we list all the factors of the numbers involved. The factors of a number are all the whole numbers that divide it evenly. Once we have the lists of factors, we compare them to find the greatest number that appears in both lists.
The GCF helps in simplifying fractions, determining ratios, and solving problems involving divisibility. In some cases, using the prime factorization method (where you break the numbers down into their prime factors) can be more efficient. However, for smaller numbers, listing the factors is straightforward and effective.
