To determine which of the following could be an integer, we will analyze each option step by step.
### Step 1: Average of 2 consecutive integers
Let the two consecutive integers be \( x \) and \( x + 1 \).
- **Calculation**:
\[
\text{Average} = \frac{x + (x + 1)}{2} = \frac{2x + 1}{2} = x + \frac{1}{2}
\]
- **Conclusion**: Since \( x \) is an integer, \( x + \frac{1}{2} \) is not an integer. Thus, the average of 2 consecutive integers cannot be an integer.
### Step 2: Average of 3 consecutive integers
Let the three consecutive integers be \( x \), \( x + 1 \), and \( x + 2 \).
- **Calculation**:
\[
\text{Average} = \frac{x + (x + 1) + (x + 2)}{3} = \frac{3x + 3}{3} = x + 1
\]
- **Conclusion**: Since \( x \) is an integer, \( x + 1 \) is also an integer. Thus, the average of 3 consecutive integers can be an integer.
### Step 3: Average of 4 consecutive integers
Let the four consecutive integers be \( x \), \( x + 1 \), \( x + 2 \), and \( x + 3 \).
- **Calculation**:
\[
\text{Average} = \frac{x + (x + 1) + (x + 2) + (x + 3)}{4} = \frac{4x + 6}{4} = x + \frac{3}{2}
\]
- **Conclusion**: Since \( x \) is an integer, \( x + \frac{3}{2} \) is not an integer. Thus, the average of 4 consecutive integers cannot be an integer.
### Step 4: Average of 6 consecutive integers
Let the six consecutive integers be \( x \), \( x + 1 \), \( x + 2 \), \( x + 3 \), \( x + 4 \), and \( x + 5 \).
- **Calculation**:
\[
\text{Average} = \frac{x + (x + 1) + (x + 2) + (x + 3) + (x + 4) + (x + 5)}{6} = \frac{6x + 15}{6} = x + \frac{5}{2}
\]
- **Conclusion**: Since \( x \) is an integer, \( x + \frac{5}{2} \) is not an integer. Thus, the average of 6 consecutive integers cannot be an integer.
### Final Conclusion
Among the options analyzed:
- The average of 2 consecutive integers: **Not an integer**
- The average of 3 consecutive integers: **Is an integer**
- The average of 4 consecutive integers: **Not an integer**
- The average of 6 consecutive integers: **Not an integer**
Thus, the only option that could be an integer is **Option 2: the average of 3 consecutive integers**.
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