What is the greatest common factor (GCF) of 80 and 50?
A. 25
B. 10
C. 5
D. 400
The correct answer and explanation is:
The correct answer is B. 10.
To find the greatest common factor (GCF) of 80 and 50, we need to identify the largest number that divides both 80 and 50 evenly.
Step 1: Find the prime factorization of each number
- 80 can be factored as follows:
- 80 ÷ 2 = 40
- 40 ÷ 2 = 20
- 20 ÷ 2 = 10
- 10 ÷ 2 = 5
- 5 ÷ 5 = 1
- Therefore, the prime factorization of 80 is 24×52^4 \times 5.
- 50 can be factored as follows:
- 50 ÷ 2 = 25
- 25 ÷ 5 = 5
- 5 ÷ 5 = 1
- Therefore, the prime factorization of 50 is 2×522 \times 5^2.
Step 2: Identify the common factors
From the prime factorizations:
- The prime factors of 80 are 24×52^4 \times 5.
- The prime factors of 50 are 2×522 \times 5^2.
The common factors between 80 and 50 are 2 and 5. We take the smallest power of each common prime factor:
- For 22, the smallest power is 212^1.
- For 55, the smallest power is 515^1.
Step 3: Multiply the common factors
The greatest common factor is the product of these common factors:
- 21×51=2×5=102^1 \times 5^1 = 2 \times 5 = 10.
Thus, the greatest common factor of 80 and 50 is 10.
Conclusion:
The greatest common factor of 80 and 50 is 10, which is option B. The GCF is the largest number that can evenly divide both numbers, and through prime factorization, we have determined that 10 is the highest such number.