Finding the Different Number Pair: Reasoning

In this question, we are given four pairs of numbers. Our task is to find the pair that does not follow the same rule or pattern as the other three pairs. The note specifies that operations should be performed on the whole numbers, not on their individual digits.

Analyzing Number Pair Patterns

We need to look for a mathematical relationship between the first number and the second number in each given pair. Let's examine each pair to see if we can identify a consistent rule.

• Pair 1: 7 : 354
• Pair 2: 5 : 136
• Pair 3: 9 : 742
• Pair 4: 8 : 523

Let's test some common relationships, such as powers of the first number, potentially with an addition or subtraction.

Checking the Cube Relationship

Let's try cubing the first number (\(n^3\)) and see if the second number (\(N\)) is related to the result.

• For 7 : 354, the first number is \(n=7\). Let's calculate \(7^3\): \[7^3 = 7 \times 7 \times 7 = 49 \times 7 = 343\] The second number is 354. The difference is \(354 - 343 = 11\). So, the relationship seems to be \(n^3 + 11\).
• For 5 : 136, the first number is \(n=5\). Let's calculate \(5^3\): \[5^3 = 5 \times 5 \times 5 = 25 \times 5 = 125\] The second number is 136. The difference is \(136 - 125 = 11\). This pair also fits the \(n^3 + 11\) relationship.
• For 9 : 742, the first number is \(n=9\). Let's calculate \(9^3\): \[9^3 = 9 \times 9 \times 9 = 81 \times 9 = 729\] The second number is 742. The difference is \(742 - 729 = 13\). This pair fits the \(n^3 + 13\) relationship, which is different from the previous two.
• For 8 : 523, the first number is \(n=8\). Let's calculate \(8^3\): \[8^3 = 8 \times 8 \times 8 = 64 \times 8 = 512\] The second number is 523. The difference is \(523 - 512 = 11\). This pair also fits the \(n^3 + 11\) relationship.

Summarizing the Number Pair Patterns

Let's put the findings into a table to clearly see the pattern for each number pair.

As the table shows, three of the number pairs (7:354, 5:136, and 8:523) follow the pattern where the second number is obtained by cubing the first number and adding 11 (\(n^3 + 11\)). The number pair 9:742 follows a different pattern, where the second number is obtained by cubing the first number and adding 13 (\(n^3 + 13\)).

Conclusion: Identifying the Different Pair

Based on the consistent pattern observed in three of the pairs, the number pair that is different is 9 : 742 because it does not follow the \(n^3 + 11\) rule. It follows a different rule, \(n^3 + 13\).

Revision Table: Key Concepts for Number Analogy

Reviewing key concepts helps reinforce understanding of number analogy questions.

• Look for common mathematical operations: addition, subtraction, multiplication, division.
• Check for powers: squares, cubes, etc.
• Consider combinations of operations (e.g., \(n^2+k\), \(n^3-k\), \(an+b\)).
• Test the identified pattern against all given pairs.
• The pair that deviates from the pattern common to the majority is the different one.

Additional Information: Logical Reasoning Patterns

Logical reasoning questions often involve identifying various types of patterns. Beyond basic arithmetic and powers, patterns can include:

• Prime numbers, composite numbers, odd/even numbers.
• Specific sequences (Fibonacci, arithmetic progression, geometric progression).
• Digit-based operations (though excluded by the note in this specific question).
• Positional value or alphabetical position relationships (in letter/number analogies).
• Combination of multiple rules.

Practicing different types of pattern recognition helps improve logical reasoning skills for competitive exams.