In this numerical odd-man-out question, four numbers are given: 110, 140, 154 and 198. Three of these numbers are divisible by 11, while one number is not a multiple of 11. Which number is the odd one out on the basis of divisibility by 11?

Introduction / Context:
This is an aptitude odd-man-out question based on divisibility properties of numbers. The numbers 110, 140, 154 and 198 are given, and you must identify which number does not share a particular factor with the others. Specifically, three of the numbers are multiples of 11, while one is not. Questions of this type check how quickly you can recognize common factors and apply basic divisibility rules, which are very important in competitive exams and number reasoning sections.

Given Data / Assumptions:
The given numbers are 110, 140, 154 and 198.
We suspect that a common factor, such as 11, is present in three numbers.
Exactly one number is not divisible by 11 and hence is the odd one out.
We work only with integer divisibility, not approximate decimal division.

Concept / Approach:
The main concept here is divisibility by 11. Although there is a specific digit-based rule for 11, in many cases multiplication knowledge is enough. If a number can be expressed as 11 * n for some integer n, then the number is divisible by 11. By factoring each given number or quickly checking division, we can determine which ones are multiples of 11. The odd-man-out is the number that does not have 11 as a factor.

Step-by-Step Solution:
Step 1: Factor 110. We can write 110 = 11 * 10, so 110 is clearly divisible by 11. Step 2: Factor 154. We can write 154 = 11 * 14, so 154 is also divisible by 11. Step 3: Factor 198. We can write 198 = 11 * 18, which shows that 198 is a multiple of 11. Step 4: Check 140. Attempting 140 / 11 gives 12 remainder 8, which is not an integer. Therefore, 140 is not divisible by 11. Step 5: Conclude that three numbers (110, 154 and 198) are multiples of 11, and 140 is not. Hence 140 is the odd one out.
Verification / Alternative check:
As an alternative, you can use the known multiplication table of 11: 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143, 154, 165, 176, 187, 198 and so on. From this list, you can see that 110, 154 and 198 all appear as multiples of 11, but 140 is missing. This confirms the earlier factorization-based conclusion without requiring full long division.

Why Other Options Are Wrong:
110 is 11 * 10, so it clearly has 11 as a factor and matches the main pattern, so it is not the odd one.
154 is 11 * 14, another clear multiple of 11, and therefore fits with the majority group.
198 is equal to 11 * 18 and so is also divisible by 11, staying consistent with the key property.
140 alone is not divisible by 11 and therefore does not share the common factor property with the others.

Common Pitfalls:
A common mistake is to look for simpler patterns like parity (even or odd) or last digit similarities. In this set all numbers are even, so parity does not help to distinguish any number. Another mistake is to stop after checking just one or two options rather than all four. In exams, it is better to systematically check each number against the suspected factor, especially when the factor is small. Recognizing table patterns, such as the table of 11, can save time and avoid careless errors.

Final Answer:
The number that is not divisible by 11 and is therefore the odd one out is 140.