Identifying the Odd Number Pair Logic

The question asks us to identify the number pair among the given options that does not follow the same relationship or logic as the others. We are given four pairs of numbers, separated by a dash (–). The rule is that operations should be performed on the whole numbers themselves, not on their individual digits.

Let's examine each number pair to find the underlying relationship between the number on the left and the number on the right.

Analyzing the Number Pairs

Option 1: 11 – 1331

Let's see if there's a common mathematical operation relating 11 to 1331. Consider the cube of 11: $11^3 = 11 \times 11 \times 11$ First, calculate $11 \times 11 = 121$. Then, calculate $121 \times 11$: $121 \times 11 = (100 + 20 + 1) \times 11 = 100 \times 11 + 20 \times 11 + 1 \times 11 = 1100 + 220 + 11 = 1331$. So, $11^3 = 1331$. The pair 11 – 1331 follows the rule where the second number is the cube of the first number.

Option 2: 12 – 1728

Let's check if the same logic applies here. Consider the cube of 12: $12^3 = 12 \times 12 \times 12$ First, calculate $12 \times 12 = 144$. Then, calculate $144 \times 12$: $144 \times 12 = (144 \times 10) + (144 \times 2) = 1440 + 288 = 1728$. So, $12^3 = 1728$. The pair 12 – 1728 also follows the rule where the second number is the cube of the first number.

Option 3: 16 – 4086

Let's apply the potential cube logic to this pair. Consider the cube of 16: $16^3 = 16 \times 16 \times 16$ First, calculate $16 \times 16 = 256$. Then, calculate $256 \times 16$: $256 \times 16 = (256 \times 10) + (256 \times 6)$ $2560 + 1536 = 4096$. So, $16^3 = 4096$. The second number in the pair is 4086, which is not equal to 4096. This pair does not follow the rule where the second number is the cube of the first number.

Option 4: 13 – 2197

Let's check this pair against the cube logic. Consider the cube of 13: $13^3 = 13 \times 13 \times 13$ First, calculate $13 \times 13 = 169$. Then, calculate $169 \times 13$: $169 \times 13 = (169 \times 10) + (169 \times 3) = 1690 + 507 = 2197$. So, $13^3 = 2197$. The pair 13 – 2197 follows the rule where the second number is the cube of the first number.

Conclusion: Finding the Odd One Out

Based on our analysis, three of the four number pairs follow the same logic: the second number is the cube of the first number. Option 1: $11^3 = 1331$ (Follows logic) Option 2: $12^3 = 1728$ (Follows logic) Option 3: $16^3 = 4096$, but the pair is 16 – 4086 (Does not follow logic) Option 4: $13^3 = 2197$ (Follows logic)

The pair 16 – 4086 is the odd one out because the second number (4086) is not the cube of the first number (16), whereas in the other three pairs, the second number is indeed the cube of the first number.

The odd pair is 16 – 4086.

Revision Table: Number Pairs & Logic

Here is a quick summary of the analysis:

Additional Information: Number Series and Patterns

Odd one out questions involving numbers often rely on identifying a pattern or rule that applies to a set of numbers or pairs, with one element not fitting the rule. Common patterns include:

• Arithmetic progression (adding/subtracting a constant)
• Geometric progression (multiplying/dividing by a constant)
• Squares of numbers ($n^2$)
• Cubes of numbers ($n^3$)
• Other powers of numbers ($n^k$)
• Prime numbers
• Composite numbers
• Addition, subtraction, multiplication, or division with a constant or based on the number itself
• Sum or product of digits (though excluded by the rule in this specific question)
• Alternating patterns

To solve such questions, it's helpful to:

1. Examine the relationship between the numbers in each pair (if pairs are given).
2. Look for simple arithmetic operations first (addition, subtraction, multiplication, division).
3. Consider squares, cubes, and other powers.
4. Check for prime or composite number properties.
5. Test the identified pattern on all options to find the one that doesn't fit.

Practice with different types of number series and pattern recognition problems improves the ability to quickly spot the underlying logic.