Find the missing digit using division equality: 2?63 ÷ 11 = 233. Determine the digit replacing the question mark.

Introduction / Context:
Here a four-digit number has one missing digit. You are told the exact quotient when dividing by 11. The task is to recover the unknown digit by reversing the division, a staple skill in number-system puzzles.

Given Data / Assumptions:

• Number form: 2?63.
• 2?63 / 11 = 233.
• All values are integers; division is exact with no remainder.

Concept / Approach:
If N / 11 = 233, then N = 233 * 11. Compute the product, then compare digits with the pattern 2?63 to read off the missing digit. This avoids long division and reduces computational effort.

Step-by-Step Solution:
Compute 233 * 11 = 233 * (10 + 1) = 2330 + 233 = 2563.Match 2563 with 2?63: clearly, ? = 5.Therefore the missing digit is 5.

Verification / Alternative check:
Divide 2563 by 11: 2563 / 11 = 233 exactly, confirming correctness and aligning with the given quotient.

Why Other Options Are Wrong:
Digits 4, 3, or 6 would produce 2463, 2363, or 2663, none of which are multiples of 11 equaling 233.

Common Pitfalls:
Trying to apply the alternating-sum test for divisibility by 11 to guess the digit, which can be slower than simply multiplying 233 * 11 for this exact quotient scenario.

Final Answer:
5