- Identify the prime factorizations: \(16 = 2 \times 2 \times 2 \times 2\) and \(24 = 2 \times 2 \times 2 \times 3\).
- Determine common factors: The shared prime factor is 2, and the minimum number of times it appears in both factorizations is 3 times.
- Calculate the GCF: Multiplying the common factors gives \(2 \times 2 \times 2\).
- Select the matching option: The greatest common factor is \(\boxed{a}\).
Explanation
- Analyzing the prime factorizations
To find the greatest common factor (GCF) of two numbers using their prime factorizations, we must identify the prime factors that both numbers share. The GCF is the product of these common prime factors, each taken at the lowest exponent found in the prime factorizations of both numbers.
The prime factorization of \(16\) is \(2 \times 2 \times 2 \times 2\), which can be written as \(2^4\).
The prime factorization of \(24\) is \(2 \times 2 \times 2 \times 3\), which can be written as \(2^3 \times 3^1\).
- Determining the greatest common factor
We identify the common prime factors in both \(16\) and \(24\). The only prime factor present in both numbers is \(2\).
Next, we look at the exponents of this common factor:
- For \(16\), the prime factor \(2\) appears \(4\) times (i.e., \(2^4\)).
- For \(24\), the prime factor \(2\) appears \(3\) times (i.e., \(2^3\)).
The greatest common factor is formed by taking the common prime factor raised to the minimum exponent found in either factorization. In this case, the minimum exponent for the prime factor \(2\) is \(3\). Therefore, the greatest common factor is \(2^3 = 2 \times 2 \times 2\).
- Selecting the correct option
Comparing our result \(2 \times 2 \times 2\) with the given options:
A\(2 \times 2 \times 2\)
B\(2 \times 2 \times 2 \times 2 \times 3\)
C\(2\)
Option 'a' matches our calculated greatest common factor. Thus, the greatest common factor of \(16\) and \(24\) is \(2 \times 2 \times 2 = 8\). The correct choice is a.
Examples
Imagine you are organizing a school fundraiser and have two different batches of snacks: 16 granola bars and 24 fruit strips. You want to create identical gift bags using all the snacks, with no items left over. To find the largest number of identical bags you can make, you need to find the greatest common factor (GCF) of 16 and 24. Since the GCF is 8, you can create 8 identical bags, each containing \(16 \div 8 = 2\) granola bars and \(24 \div 8 = 3\) fruit strips. This ensures that every bag is exactly the same and no snacks are wasted.