What is the greatest common factor (GCF) of 32 and 48? A. 96 B. 16 C. 8 D. 32
The correct answer and explanation is:
Correct Answer: B. 16
To find the Greatest Common Factor (GCF) of 32 and 48, you need to determine the largest number that can divide both numbers exactly, with no remainder.
Step-by-step process:
- Prime Factorization
- 32 = 2 × 2 × 2 × 2 × 2 = 2⁵
- 48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3
- Compare Common Prime Factors
The prime factors they both share are powers of 2.- 32 has five 2s
- 48 has four 2s
So, they both share four 2s, or 2⁴.
- Multiply Shared Factors
2⁴ = 2 × 2 × 2 × 2 = 16
So, the greatest number that divides both 32 and 48 is 16.
Why not the other options?
- A. 96: 96 is larger than both 32 and 48. A common factor must be less than or equal to the smallest number, which is 32. So 96 cannot be the GCF.
- C. 8: 8 is a common factor of 32 and 48, but it is not the greatest. 16 is larger and still divides both numbers.
- D. 32: 32 is a factor of 32 but not of 48. 48 ÷ 32 = 1.5, which is not a whole number.
Conclusion
The GCF is the largest number that can divide both 32 and 48 without leaving a remainder. The correct answer is 16 because it is the highest number that divides both evenly. It plays an important role in simplifying fractions and solving ratio problems. Understanding how to find the GCF helps strengthen skills in number theory and arithmetic.