What is the greatest common factor for 96, 72 48?
The correct answer and explanation is:
To find the greatest common factor (GCF) of 96, 72, and 48, start by finding the prime factorization of each number.
- Prime factorization of 96:
- 96 is divisible by 2: 96 ÷ 2 = 48
- 48 ÷ 2 = 24
- 24 ÷ 2 = 12
- 12 ÷ 2 = 6
- 6 ÷ 2 = 3 (now 3 is a prime number, so stop here)
- Prime factorization: 96=25×396 = 2^5 \times 3
- Prime factorization of 72:
- 72 ÷ 2 = 36
- 36 ÷ 2 = 18
- 18 ÷ 2 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
- Prime factorization: 72=23×3272 = 2^3 \times 3^2
- Prime factorization of 48:
- 48 ÷ 2 = 24
- 24 ÷ 2 = 12
- 12 ÷ 2 = 6
- 6 ÷ 2 = 3
- Prime factorization: 48=24×348 = 2^4 \times 3
Now, to find the GCF, identify the common factors among the three numbers. Look at the powers of 2 and 3:
- The lowest power of 2 is 232^3 (appears in 72).
- The lowest power of 3 is 313^1 (appears in both 96 and 48).
Therefore, the GCF is the product of the lowest powers of the common factors: GCF=23×3=8×3=24\text{GCF} = 2^3 \times 3 = 8 \times 3 = 24
So, the greatest common factor of 96, 72, and 48 is 24.
This method works by finding the prime factorizations of each number, comparing the common factors, and taking the lowest power of each common factor. The GCF is the product of these lowest powers. This method ensures that you are finding the largest number that divides all the given numbers evenly.