What is the greatest common factor of 28, 48, and 64?
The correct answer and explanation is:
Correct Answer: The greatest common factor (GCF) of 28, 48, and 64 is 4.
Explanation:
The greatest common factor (GCF) is the largest positive number that divides exactly into each of the given numbers without leaving a remainder. To find the GCF of 28, 48, and 64, the process starts by identifying the prime factors of each number.
- Prime factorization of 28:
28 = 2 × 2 × 7 = 2² × 7 - Prime factorization of 48:
48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3 - Prime factorization of 64:
64 = 2 × 2 × 2 × 2 × 2 × 2 = 2⁶
Now, observe the prime factors common to all three numbers.
The only prime factor common to all three numbers is 2.
- 28 has two 2s (2²)
- 48 has four 2s (2⁴)
- 64 has six 2s (2⁶)
To find the GCF, take the lowest exponent of the common prime factor:
- For 2, the lowest exponent is 2 (from 28)
Therefore, the GCF is 2² = 4
This means 4 is the largest number that can divide 28, 48, and 64 evenly.
Why this matters:
Finding the GCF is useful in simplifying fractions, solving problems involving ratios, and determining common denominators. It is a key concept in number theory and arithmetic. By breaking each number down into its prime components, it becomes easy to identify shared factors and find the greatest one they have in common. This method is accurate and systematic, and it works even with larger numbers.