To solve the problem, we need to find the number of even factors for the even positive integers 48 and 122, excluding the number itself. Let's break this down step by step.
### Step 1: Find the even factors of 48
1. **Prime Factorization of 48**:
- 48 can be expressed as \( 48 = 2^4 \times 3^1 \).
2. **Finding the total number of factors**:
- The formula to find the total number of factors from the prime factorization \( p_1^{e_1} \times p_2^{e_2} \) is \( (e_1 + 1)(e_2 + 1) \).
- For 48, the total number of factors is \( (4 + 1)(1 + 1) = 5 \times 2 = 10 \).
3. **Finding the even factors**:
- Since we are interested in even factors, we can consider the factorization without the odd part.
- The even factors of 48 can be calculated by taking at least one factor of 2 from the prime factorization.
- The even factors can be derived from \( 2^1 \times 2^2 \times 3^0 \) to \( 2^4 \times 3^1 \), which gives us:
- \( 2^1 \): 2, 6, 12, 24, 48
- \( 2^2 \): 4, 8, 16, 48
- \( 2^3 \): 8, 16, 48
- \( 2^4 \): 16, 48
- The even factors of 48 are: 2, 4, 6, 8, 12, 16, 24, 48.
- Excluding 48, we have: 2, 4, 6, 8, 12, 16, 24.
- Thus, the number of even factors of 48 excluding itself is **7**.
### Step 2: Find the even factors of 122
1. **Prime Factorization of 122**:
- 122 can be expressed as \( 122 = 2^1 \times 61^1 \).
2. **Finding the total number of factors**:
- The total number of factors is \( (1 + 1)(1 + 1) = 2 \times 2 = 4 \).
3. **Finding the even factors**:
- The even factors of 122 are derived from the factorization that includes at least one factor of 2.
- The even factors of 122 are: 2, 61, 122.
- Excluding 122, we have only one even factor which is **2**.
### Conclusion
- The number of even factors of 48 (excluding 48 itself) is **7**.
- The number of even factors of 122 (excluding 122 itself) is **1**.
### Final Comparison
- Column A (star of 48) has 7 even factors.
- Column B (star of 122) has 1 even factor.
- Therefore, Column A is greater than Column B.
### Answer
Column A is larger than Column B.
