Solve for the missing factor: Find the exact integer that satisfies ? × 11 = 555555.

Introduction / Context:
This is a reverse multiplication problem. Instead of multiplying, you are asked to divide a patterned number by 11 to recover the exact integer factor. Recognizing repeating-digit structures can speed up calculation.

Given Data / Assumptions:

• Equation: ? × 11 = 555555.
• All values are integers.
• No remainder is implied; an exact integer solution is expected.

Concept / Approach:
Since 11 × 50505 = 50505 × (10 + 1) = 505050 + 50505 = 555555, the clean repeating pattern suggests 50505 is the factor. This avoids long division and relies on the structure of base-10 multiplication by 11.

Step-by-Step Solution:
Compute 50505 × 10 = 505050.Add 50505 (for the +1 part): 505050 + 50505 = 555555.Hence the missing factor is 50505.

Verification / Alternative check:
Divide directly: 555555 ÷ 11 = 50505. The quotient is an integer with no remainder, confirming the choice.

Why Other Options Are Wrong:
505, 5050, 5005 produce significantly smaller products with 11 and cannot reach 555555.

Common Pitfalls:
Guessing small factors due to the compact look of 11; forgetting that multiplying by 11 roughly scales by ten-plus-one, so the factor should have a similar number of digits as the quotient pattern.

Final Answer:
50505