Understanding the Problem
To find the greatest common factor (GCF) of 126 and 196, we can use the prime factorization method.

Step 1: Prime Factorization
- **Prime Factorization of 126:**
- 126 can be divided by 2 (the smallest prime number):
- 126 ÷ 2 = 63
- Next, factor 63:
- 63 ÷ 3 = 21
- 21 ÷ 3 = 7
- So, the prime factors of 126 are:
- **2 x 3² x 7**
- **Prime Factorization of 196:**
- 196 can also be divided by 2:
- 196 ÷ 2 = 98
- 98 ÷ 2 = 49
- Next, factor 49:
- 49 ÷ 7 = 7
- Thus, the prime factors of 196 are:
- **2² x 7²**

Step 2: Identifying Common Factors
- The prime factors of 126 are:
- **2, 3², 7**
- The prime factors of 196 are:
- **2², 7²**
- The common prime factors are:
- **2 and 7**

Step 3: Calculating the GCF
- To find the GCF, take the lowest power of each common prime factor:
- For **2**: Minimum power is 1 (from 2¹)
- For **7**: Minimum power is 1 (from 7¹)
- Therefore, the GCF is:
- **GCF = 2¹ x 7¹ = 2 x 7 = 14**

Final Result
The greatest common factor of 126 and 196 is **14**.