Question

What is the greatest common factor of 24 and 32?

Answer 1

To find the greatest common factor (GCF) of 24 and 32, we can first find the prime factorization of each number.

For 24:
- 24 can be divided by 2: \( 24 \div 2 = 12 \)
- 12 can also be divided by 2: \( 12 \div 2 = 6 \)
- 6 can again be divided by 2: \( 6 \div 2 = 3 \)
- 3 is a prime number.

So, the prime factorization of 24 is:
\[
24 = 2^3 \times 3^1
\]

For 32:
- 32 can be divided by 2: \( 32 \div 2 = 16 \)
- 16 can also be divided by 2: \( 16 \div 2 = 8 \)
- 8 can again be divided by 2: \( 8 \div 2 = 4 \)
- 4 can again be divided by 2: \( 4 \div 2 = 2 \)
- 2 is a prime number.

So, the prime factorization of 32 is:
\[
32 = 2^5
\]

Now, we find the common factors using the prime factorization:
- The common prime factor is \( 2 \).
- The lowest power of the common prime factor \( 2 \) is \( 2^3 \) (since \( 2^3 \) from 24 and \( 2^5 \) from 32).

Therefore, the GCF of 24 and 32 is:
\[
2^3 = 8
\]

Thus, the greatest common factor of 24 and 32 is \( \boxed{8} \).