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To solve the problem, let's apply the rule for divisibility by 11. 

For a number to be divisible by 11, the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions must be either 0 or a multiple of 11.

The number given is \( X9257Y684 \).

### Step 1: Identify the digits in odd and even positions.
- Odd positions: \( X, 2, 5, Y, 8 \)
- Even positions: \( 9, 5, 7, 6, 4 \)

### Step 2: Calculate the sums.
- Sum of digits in odd positions: \( X + 2 + 5 + Y + 8 = X + Y + 15 \)
- Sum of digits in even positions: \( 9 + 5 + 7 + 6 + 4 = 31 \)

### Step 3: Apply the divisibility rule.
For divisibility by 11:
\[
(X + Y + 15) - 31 = X + Y - 16 
\]
This difference \( (X + Y - 16) \) must be 0 or a multiple of 11. Thus, we have:
\[
X + Y - 16 = 0 \quad \text{or} \quad X + Y - 16 = \pm 11
\]
\[
X + Y = 16 \quad \text{or} \quad X + Y = 27 \quad \text{or} \quad X + Y = 5
\]

### Step 4: Calculate \( X - Y \).
We need to minimize \( X - Y \).

1. For \( X + Y = 16 \), possible pairs \((X, Y)\) are:
   - \( (8, 8) \) → \( X - Y = 0 \) (smallest)
   - \( (9, 7) \) → \( X - Y = 2 \)
   - \( (7, 9) \) → \( X - Y = -2 \)

2. For \( X + Y = 27 \), possible pairs \((X, Y)\) are:
   - \( (9, 9) \) → \( X - Y = 0 \) 

3. For \( X + Y = 5 \), possible pairs \((X, Y)\) are:
   - \( (3, 2) \) → \( X - Y = 1 \)
   - \( (4, 1) \) → \( X - Y = 3 \)
   - \( (5, 0) \) → \( X - Y = 5 \)

### Step 5: Determine the minimum value of \( X - Y \).

From the options, the least value of \( X - Y \) is \( -2 \), achieved when \( X = 7 \) and \( Y = 9 \).

**Final Answer: The least value of \( X - Y \) is \(-2\).**

Would you like more details or have any questions?

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**Here are five related questions:**

1. What is the rule of divisibility for 7 and how can it be applied?
2. How can we determine if a number is divisible by 13 using its digits?
3. What is the general rule of divisibility for any prime number?
4. Can you explain how to find the largest value of \( X - Y \) given similar conditions?
5. What other mathematical properties can be derived from the sum of digits of a number?

**Tip:** When working with divisibility rules, always check for multiple possible sums to ensure the correct result.

If the number X9257Y684 is divisible by 11, then what is the least value of X - Y?

Learn how to determine the least value of X - Y for the number X9257Y684 to be divisible by 11 using the divisibility rule. This problem covers key number theory concepts and is suitable for students in Grades 9-10.

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