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To solve this problem, we need to express 55 as the sum of \( n \) consecutive integers and then find the difference between the maximum and minimum values of \( n \).

### Step-by-Step Solution

1. **Define Consecutive Integers:**
   Suppose we have \( n \) consecutive integers starting from some integer \( a \). The sequence of these integers would be:
   \[
   a, a+1, a+2, \ldots, a+(n-1)
   \]

2. **Sum of \( n \) Consecutive Integers:**
   The sum of \( n \) consecutive integers starting from \( a \) can be represented as:
   \[
   S = a + (a + 1) + (a + 2) + \ldots + (a + n - 1)
   \]
   Using the formula for the sum of an arithmetic sequence, we have:
   \[
   S = \frac{n}{2} \times (2a + n - 1)
   \]
   We are given that this sum \( S = 55 \). So,
   \[
   \frac{n}{2} \times (2a + n - 1) = 55
   \]

3. **Solving for \( n \) and \( a \):**
   Rearranging the equation:
   \[
   n \times (2a + n - 1) = 110
   \]
   We need to find pairs of \( n \) and \( a \) such that this equation holds and results in consecutive integers.

4. **Testing Possible Values of \( n \):**
   We’ll try different values of \( n \) that divide 110 to see which satisfy the equation.

   - For \( n = 10 \):
     \[
     2a + 10 - 1 = \frac{110}{10} = 11
     \]
     \[
     2a = 11 - 9 = 2 \Rightarrow a = 1
     \]
     So, the integers are \( 1, 2, 3, \ldots, 10 \), and the maximum and minimum are 10 and 1, respectively.

5. **Calculate \( \max(n) - \min(n) \):**
   \[
   \max(n) - \min(n) = 10 - 1 = 9
   \]

### Final Answer
The value of \( \max(n) - \min(n) \) is **9**.

Would you like further details on any specific part of this solution?

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### Related Questions

1. How do we determine if a number can be written as the sum of consecutive integers?
2. What other values of \( n \) would work if the sum were a different number?
3. How can we generalize the approach to any number \( S \) for finding consecutive integers?
4. Can we use this method to find sequences of consecutive integers for sums that aren't integers?
5. Why does the difference between \( \max(n) \) and \( \min(n) \) yield a consistent result in problems like these?

### Tip
When dealing with consecutive integers summing to a particular number, break down the problem by dividing the sum by possible values of \( n \), which helps in isolating the initial integer \( a \).

55 can be written as the sum of n consecutive integers. Find the value of max(n) - min(n).

Learn how to express 55 as the sum of n consecutive integers and find the difference between the maximum and minimum values of n. This problem covers key arithmetic concepts and is suitable for students in Grades 8-10.

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