x is both an integer and greater than 1.integer factor of x not equal to 1`" "`Let[x] stand for the smallest positive integer factor of x not equal to 1
`{:("Column A", " ","Column B"),([x],,[x^3]):}`

Options

A. If column A is larger
B. If column B is larger
C. If the columns are equal
D. If there is not enough information to decide

Answer

To solve the problem, we need to compare the values in Column A and Column B based on the given definition of the smallest positive integer factor of \( x \) that is not equal to 1.

### Step 1: Understand the Definitions
- Let \( [x] \) represent the smallest positive integer factor of \( x \) that is not equal to 1.
- Column A is \( [x] \).
- Column B is \( x^3 \).

### Step 2: Analyze the Values of \( x \)
Since \( x \) is an integer greater than 1, we can test a few integer values for \( x \) to see how \( [x] \) and \( x^3 \) compare.

### Step 3: Test with \( x = 2 \)
1. **Calculate \( [2] \)**:
- The smallest positive integer factor of 2 that is not equal to 1 is 2.
- Therefore, \( [2] = 2 \).

2. **Calculate \( 2^3 \)**:
- \( 2^3 = 8 \).

3. **Compare**:
- Column A: \( [2] = 2 \)
- Column B: \( 2^3 = 8 \)
- Conclusion: Column A < Column B.

### Step 4: Test with \( x = 3 \)
1. **Calculate \( [3] \)**:
- The smallest positive integer factor of 3 that is not equal to 1 is 3.
- Therefore, \( [3] = 3 \).

2. **Calculate \( 3^3 \)**:
- \( 3^3 = 27 \).

3. **Compare**:
- Column A: \( [3] = 3 \)
- Column B: \( 3^3 = 27 \)
- Conclusion: Column A < Column B.

### Step 5: Test with \( x = 4 \)
1. **Calculate \( [4] \)**:
- The smallest positive integer factor of 4 that is not equal to 1 is 2.
- Therefore, \( [4] = 2 \).

2. **Calculate \( 4^3 \)**:
- \( 4^3 = 64 \).

3. **Compare**:
- Column A: \( [4] = 2 \)
- Column B: \( 4^3 = 64 \)
- Conclusion: Column A < Column B.

### Step 6: General Observation
From the tests conducted:
- For \( x = 2 \), \( [x] < x^3 \).
- For \( x = 3 \), \( [x] < x^3 \).
- For \( x = 4 \), \( [x] < x^3 \).

### Conclusion
In all cases tested, \( [x] < x^3 \). Therefore, we can conclude that for any integer \( x > 1 \), Column A is always less than Column B.

### Final Answer
The correct answer is that Column B is greater than Column A for all integers \( x > 1 \).

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