What is the greatest common factor of 24 and 64? a. 8 b. 4 c. 12 d. 36
The Correct Answer and Explanation is:
The greatest common factor (GCF) of 24 and 64 is a. 8.
Explanation:
To find the greatest common factor, we need to first determine the prime factorizations of both numbers and then identify the highest number that divides both.
1. Prime factorization of 24:
- 24 can be divided by 2 (the smallest prime number):
24 ÷ 2 = 12
12 ÷ 2 = 6
6 ÷ 2 = 3
3 ÷ 3 = 1
So, the prime factorization of 24 is:
2 × 2 × 2 × 3 or 2³ × 3.
2. Prime factorization of 64:
- 64 is also divisible by 2:
64 ÷ 2 = 32
32 ÷ 2 = 16
16 ÷ 2 = 8
8 ÷ 2 = 4
4 ÷ 2 = 2
2 ÷ 2 = 1
So, the prime factorization of 64 is:
2 × 2 × 2 × 2 × 2 × 2 or 2⁶.
3. Identify the common factors:
- The prime factorizations are:
24 = 2³ × 3
64 = 2⁶ - The common factor between 24 and 64 is 2³ (since both numbers have at least three factors of 2).
4. Find the greatest common factor:
- The greatest common factor is the highest power of the common prime factor, which in this case is 2³ = 8.
Thus, the greatest common factor of 24 and 64 is 8, and the correct answer is a. 8.
