Finding the Odd Number Pair Using Mathematical Logic

The question asks us to identify the number pair that does not follow the same mathematical operation or set of operations as the other pairs. We are given four number pairs, where the second number is derived from the first number using a consistent rule for three of the pairs.

Analyzing the Given Number Pairs

Let's look closely at the given options:

• 24 : 169
• 12 : 49
• 48 : 625
• 36 : 365

We need to find a relationship between the first number and the second number in each pair. Let's examine the second numbers: 169, 49, 625, and 365.

• 169 is a perfect square: $169 = 13^2$.
• 49 is a perfect square: $49 = 7^2$.
• 625 is a perfect square: $625 = 25^2$.
• 365 is not a perfect square.

This observation suggests that the rule might involve squaring a number derived from the first number. Let's investigate how the bases of the squares (13, 7, 25) relate to the corresponding first numbers (24, 12, 48).

Identifying the Mathematical Rule

Let the first number be $N_1$ and the second number be $N_2$. Based on our analysis of the perfect squares, it seems $N_2$ is the square of some value related to $N_1$. Let's test potential relationships between $N_1$ and the base of the square ($B$) for the first three pairs:

• For 24 : 169, $N_1 = 24$, $N_2 = 169 = 13^2$. The base is $B = 13$. How can we get 13 from 24?
  • $24 / 2 = 12$. $12 + 1 = 13$. Let's check this pattern.
• For 12 : 49, $N_1 = 12$, $N_2 = 49 = 7^2$. The base is $B = 7$. Using the potential pattern:
  • $12 / 2 = 6$. $6 + 1 = 7$. This matches the base. Let's square it: $(7)^2 = 49$. This matches the second number.
• For 48 : 625, $N_1 = 48$, $N_2 = 625 = 25^2$. The base is $B = 25$. Using the potential pattern:
  • $48 / 2 = 24$. $24 + 1 = 25$. This matches the base. Let's square it: $(25)^2 = 625$. This matches the second number.

The pattern that emerges is: Take the first number ($N_1$), divide it by 2, add 1, and then square the result to get the second number ($N_2$). Mathematically, the rule is $N_2 = \left(\frac{N_1}{2} + 1\right)^2$.

Testing the Rule on All Pairs

Let's apply this rule to all four number pairs:

As shown in the table, the first three number pairs (24 : 169, 12 : 49, and 48 : 625) follow the rule $\left(\frac{N_1}{2} + 1\right)^2 = N_2$. However, the fourth pair (36 : 365) does not follow this rule, as $\left(\frac{36}{2} + 1\right)^2 = 19^2 = 361$, which is not equal to 365.

Conclusion

The number pair that does not follow the same mathematical operation as the others is 36 : 365. This is the odd number-pair.

The final answer is the number pair 36 : 365.

Revision Table: Number Pair Logic

Additional Information: Solving Logic-Based Number Problems

Problems involving finding patterns in number pairs or series are common in logical reasoning and quantitative aptitude tests. Here are some tips for solving such problems:

• Look for simple arithmetic operations: Addition, subtraction, multiplication, division. Is there a constant difference or ratio?
• Consider squares and cubes: Many patterns involve squaring or cubing numbers, or adding/subtracting from squares/cubes.
• Check for relationships involving digits: Sometimes the operation is on the digits of the number (e.g., sum of digits, product of digits).
• Look for sequences: Fibonacci sequence, prime numbers, etc., might be involved.
• Test potential rules systematically: Once you hypothesize a rule, apply it to all given examples to see if it holds true.
• Eliminate options: If a rule works for several options, test the remaining options to find the one that breaks the rule.

Practice with various types of number pattern problems will help you become quicker at spotting the underlying logic.