What is the greatest common factor of 32, 44, and 48?
The correct answer and explanation is:
The greatest common factor (GCF) of 32, 44, and 48 is 4.
Explanation:
To find the GCF of these numbers, we need to follow a few key steps:
- Prime Factorization:
- For 32:
32=2532 = 2^5 - For 44:
44=22×1144 = 2^2 \times 11 - For 48:
48=24×348 = 2^4 \times 3
- For 32:
- Identify Common Factors:
To find the GCF, we look for the common prime factors in all three numbers. The only common prime factor here is 2. - Lowest Power of Common Prime Factor:
- The powers of 2 in the factorizations are:
- 32=2532 = 2^5
- 44=2244 = 2^2
- 48=2448 = 2^4
- The lowest power of 2 that appears in all three factorizations is 222^2, which equals 4.
- The powers of 2 in the factorizations are:
Thus, the GCF of 32, 44, and 48 is 4, because 4 is the largest number that divides all three numbers without a remainder.
Why It Matters:
The GCF is useful in simplifying fractions, finding common denominators, and solving problems that involve dividing things into equal parts. When working with multiple numbers, finding the GCF ensures you are working with the most reduced form of the numbers involved.