Find the GCF of 28 and 32 28 32 X GCF=
The Correct Answer and Explanation is:
To find the greatest common factor (GCF) of 28 and 32, we need to identify the largest number that divides both 28 and 32 without leaving a remainder.
Step 1: Prime Factorization
Start by performing the prime factorization of both numbers:
- 28: To find the prime factors of 28, divide by the smallest prime number (2).
28 ÷ 2 = 14
14 ÷ 2 = 7
Since 7 is a prime number, the prime factorization of 28 is:
28 = 2 × 2 × 7 (or 2² × 7) - 32: Now, for 32, divide by 2 as well.
32 ÷ 2 = 16
16 ÷ 2 = 8
8 ÷ 2 = 4
4 ÷ 2 = 2
2 ÷ 2 = 1
So, the prime factorization of 32 is:
32 = 2 × 2 × 2 × 2 × 2 (or 2⁵)
Step 2: Identify Common Factors
Now, let’s compare the prime factorizations of 28 and 32:
- 28 = 2² × 7
- 32 = 2⁵
Both numbers share the factor of 2. The smallest power of 2 that is common in both factorizations is 2².
Step 3: Calculate the GCF
The GCF is the product of the common factors. In this case, the only common factor is 2², which is 4.
Thus, the greatest common factor of 28 and 32 is 4.
Explanation
The concept of finding the GCF involves identifying the largest number that divides two or more numbers without leaving a remainder. This can be done by finding the prime factors of both numbers and selecting the common ones. When there are multiple factors in common, you select the smallest power of each common factor to ensure you get the greatest divisor. In this case, both 28 and 32 share the factor 2, and the highest power of 2 common to both is 2², so the GCF is 4.
This method is useful because it not only works for small numbers like 28 and 32 but also scales well for larger numbers or more complex factorizations.
