What digit should replace the blank so that the number 275_476 becomes exactly divisible by 11?

Introduction / Context:
This problem focuses on the divisibility rule for 11 and asks you to find a missing digit in a large number. The number 275_476 contains a blank digit that must be chosen so that the entire number is divisible by 11. Such questions are common in competitive exams to test knowledge of divisibility rules and comfort with handling multi digit numbers without full long division.

Given Data / Assumptions:

- The number has the form 275x476, where x is a single digit from 0 to 9. - The complete number must be divisible by 11. - We must determine which digit x satisfies this condition.

Concept / Approach:
The divisibility rule for 11 states that a number is divisible by 11 if and only if the difference between the sum of the digits in odd positions and the sum of the digits in even positions (counted from the left) is a multiple of 11, including 0. We apply this rule to the seven digit number 275x476, derive an expression in terms of the unknown digit x and then solve for x using simple modular reasoning.

Step-by-Step Solution:
Step 1: Write the digits of 275x476 explicitly: 2, 7, 5, x, 4, 7, 6. Step 2: Label the positions from the left: position 1 is 2, 2 is 7, 3 is 5, 4 is x, 5 is 4, 6 is 7 and 7 is 6. Step 3: Identify odd positions (1, 3, 5, 7) and even positions (2, 4, 6). Step 4: Sum the digits in odd positions: 2 + 5 + 4 + 6 = 17. Step 5: Sum the digits in even positions: 7 + x + 7 = 14 + x. Step 6: Compute the difference between these sums: 17 - (14 + x) = 3 - x. Step 7: For divisibility by 11, this difference must be a multiple of 11. The simplest possibility is 3 - x = 0, which gives x = 3. Step 8: Other multiples like 3 - x = 11 or 3 - x = -11 would lead to x = -8 or x = 14, which are not valid digits. Step 9: Therefore the only valid digit that works is x = 3.

Verification / Alternative check:
Substitute x = 3 into the number to get 2753476. Recalculate the divisibility rule. Sum of odd position digits remains 17. Sum of even position digits becomes 7 + 3 + 7 = 17. The difference is 17 - 17 = 0, which is a multiple of 11. This confirms that 2753476 is divisible by 11. No other option digit satisfies this condition without breaking the rule.

Why Other Options Are Wrong:
For x = 6, 4 or 2, the difference 3 - x becomes -3, -1 or 1 respectively, none of which are multiples of 11. Hence the resulting numbers are not divisible by 11. These values represent common incorrect guesses when the divisibility rule is not applied correctly.

Common Pitfalls:
Typical mistakes include adding all digits together instead of separating odd and even positions or miscounting the positions from the left. Sometimes students take the absolute value of the difference and compare it to 11 without allowing 0, which is also a multiple of 11. Carefully labeling positions and writing separate sums avoids these errors.

Final Answer:
The missing digit that makes 275_476 divisible by 11 is 3, which corresponds to option D.