From the numbers 8912, 3469, 5555 and 6734, identify the odd one out using a divisibility by 11 pattern.

Introduction / Context:
Odd one out questions with four digit numbers often rely on recognizable digit patterns or simple divisibility rules. In this problem, the four options look quite different at first glance, but one of them has a fundamental numerical property that the others lack. We can either notice the pattern by observation or confirm it using a divisibility test. Two simple clues are that 5555 has all digits the same and that it satisfies the test for divisibility by 11.

Given Data / Assumptions:
The numbers are 8912, 3469, 5555 and 6734. We check both digit pattern and divisibility by 11. We use the standard divisibility rule for 11 based on alternating sums of digits.

Concept / Approach:
There are two supporting concepts here. First, 5555 has all four digits identical, while the other numbers do not, which already makes it visually distinct. Second, the divisibility rule for 11 states that for a number abcd, we compute (a + c) minus (b + d). If the result is 0 or a multiple of 11, the number is divisible by 11. When we apply this rule, we find that only 5555 is divisible by 11, which gives a clear mathematical justification for it being the odd one out.

Step-by-Step Solution:
Step 1: Note that 5555 has a special pattern where all digits are 5, unlike the mixed digit patterns in 8912, 3469 and 6734. Step 2: Apply the rule for 11 to 8912. Sum of first and third digits = 8 + 1 = 9. Sum of second and fourth digits = 9 + 2 = 11. Difference = 9 - 11 = -2, not a multiple of 11, so 8912 is not divisible by 11. Step 3: For 3469, sum of first and third digits = 3 + 6 = 9. Sum of second and fourth digits = 4 + 9 = 13. Difference = 9 - 13 = -4, not a multiple of 11. Step 4: For 5555, sum of first and third digits = 5 + 5 = 10. Sum of second and fourth digits = 5 + 5 = 10. Difference = 10 - 10 = 0, which is a multiple of 11, so 5555 is divisible by 11. Step 5: For 6734, sum of first and third digits = 6 + 3 = 9. Sum of second and fourth digits = 7 + 4 = 11. Difference = 9 - 11 = -2, not a multiple of 11. Step 6: Therefore, 5555 alone is divisible by 11 and also visually distinct by having all identical digits, so it is the odd one out.

Verification / Alternative check:
We can confirm by direct division: 5555 divided by 11 equals 505, an integer, which proves divisibility. For 8912, 3469 and 6734, division by 11 does not give integer results, consistent with the digit rule. The repeated digit pattern in 5555 reinforces its uniqueness even without performing detailed calculations.

Why Other Options Are Wrong:
8912 is not the odd one out because it is not divisible by 11 and has mixed digits like 3469 and 6734. 3469 is not the odd one out because it also fails the 11 divisibility test and does not show any special digit repetition. 6734 is not the odd one out because it follows the same pattern as 8912 and 3469 with mixed digits and no divisibility by 11. 5555 stands apart both in digit pattern and divisibility, so the other three numbers form a natural group when compared against it.

Common Pitfalls:
Some students may focus only on the visual clue of repeated digits and ignore the confirmation by divisibility, which is acceptable but less rigorous. Others might wrongly assume there is a more complicated arithmetic pattern when a simple rule suffices. Developing the habit of checking straightforward divisibility rules avoids overcomplicating such questions and saves time in competitive exams.

Final Answer:
The number that is divisible by 11 and also has all digits equal, and is therefore the odd one out, is 5555.