To find the greatest common factor (GCF) using the prime factorizations of 16 and 24, we look for the common prime factors and take the lowest power of these common factors.
The prime factorization of each number is:
- 16: \(2^4\) (which means 2, 2, 2, 2)
- 24: \(2^3 \times 3^1\) (which means 2, 2, 2, 3)
The common prime factor is \(2\).
- In 16, the number of \(2\)s is \(4\) (or \(2^4\)).
- In 24, the number of \(2\)s is \(3\) (or \(2^3\)).
We take the lowest power of \(2\) common to both numbers, which is \(2^3\).
Now we convert \(2^3\) back to its numerical form:
- \(2^3 = 2 \times 2 \times 2 = 8\)
Thus, the greatest common factor (GCF) of 16 and 24 is 8.
However, based on the provided answer options, none directly mention 8, but it seems like they are working with products of the prime factors for each number. If we interpret the options as follows:
- \(2\) represents \(2^1\)
- \(2 \times 2\) represents \(2^2\)
- \(2 \times 2 \times 2\) represents \(2^3\)
- \(2 \times 2 \times 2 \times 2 \times 3\) represents \(2^4 \times 3^1\) (not relevant for GCF)
The correct choice would be \(2 \times 2 \times 2 = 8\), which corresponds to the option \(2 \times 2 \times 2\) from the given options, indicating that the GCF is \(8\).