Finding the Odd Number Pair Pattern

The question asks us to identify the number pair that does not follow the same mathematical rule as the other three pairs. We are given four pairs of numbers. The second number in each pair is obtained by applying a certain operation(s) to the first number.

Let's look at the given number pairs:

• 5 ∶ 13
• 2 ∶ 6
• 3 ∶ 9
• 6 ∶ 15

Analyzing Number Pairs to Discover the Rule

We need to find a relationship between the first and second number in each pair. Let's examine each pair and try to find a pattern. We can test common mathematical operations like addition, subtraction, multiplication, division, squares, cubes, or combinations of these.

Testing Possible Mathematical Operations

Let the first number be $x$ and the second number be $y$. We are looking for a rule $y = f(x)$ that applies to most pairs.

Let's consider simple multiplication:

• 5 ∶ 13: $13 / 5 = 2.6$
• 2 ∶ 6: $6 / 2 = 3$
• 3 ∶ 9: $9 / 3 = 3$
• 6 ∶ 15: $15 / 6 = 2.5$

This doesn't show a consistent factor for all pairs.

Let's consider multiplication and addition/subtraction. How about multiplying the first number by some value and adding/subtracting another value?

Let's try multiplying by 2 and adding/subtracting something:

• 5 ∶ 13: $5 \times 2 = 10$. To get 13, we add 3. So, $5 \times 2 + 3 = 13$.
• 2 ∶ 6: $2 \times 2 = 4$. To get 6, we add 2. So, $2 \times 2 + 2 = 6$.
• 3 ∶ 9: $3 \times 2 = 6$. To get 9, we add 3. So, $3 \times 2 + 3 = 9$.
• 6 ∶ 15: $6 \times 2 = 12$. To get 15, we add 3. So, $6 \times 2 + 3 = 15$.

Looking at these results, the operation "$x \times 2 + 3$" seems to work for the pairs (5, 13), (3, 9), and (6, 15). Let's verify this rule for these three pairs:

• For (5, 13): $5 \times 2 + 3 = 10 + 3 = 13$. This matches the second number.
• For (3, 9): $3 \times 2 + 3 = 6 + 3 = 9$. This matches the second number.
• For (6, 15): $6 \times 2 + 3 = 12 + 3 = 15$. This matches the second number.

Now let's apply this rule to the pair (2, 6):

• For (2, 6): $2 \times 2 + 3 = 4 + 3 = 7$. This result (7) does not match the second number in the pair (6).

This indicates that the pair 2 ∶ 6 does not follow the pattern "multiply the first number by 2 and add 3".

Let's consider another possible pattern that works for some pairs, multiplying by 3:

• 5 ∶ 13: $5 \times 3 = 15$ (Not 13)
• 2 ∶ 6: $2 \times 3 = 6$ (Matches)
• 3 ∶ 9: $3 \times 3 = 9$ (Matches)
• 6 ∶ 15: $6 \times 3 = 18$ (Not 15)

This pattern only works for two out of four pairs, which is not the rule followed by "all number pairs except one". The rule "multiply by 2 and add 3" works for three out of four pairs.

Identifying the Odd Number Pair

Based on our analysis, the rule that connects the first number to the second number in three of the four pairs is $y = 2x + 3$. The pair that does not follow this rule is 2 ∶ 6, where applying the rule gives $2 \times 2 + 3 = 7$, not 6.

Therefore, the odd number pair is 2 ∶ 6.

Conclusion on Odd Number Pair

The analysis confirms that the pair 2 ∶ 6 is the one that deviates from the pattern followed by the other number pairs.

Revision Table: Number Pattern Analysis

Reviewing the steps taken to find the odd number pair:

• Examined each given number pair.
• Hypothesized a possible mathematical relationship ($y = 2x + 3$).
• Tested the hypothesis on all pairs.
• Identified the pair that did not fit the relationship.

Additional Information: Number Series and Pattern Recognition

Questions like this involve pattern recognition, a common skill tested in logical reasoning and quantitative aptitude. These patterns can be simple arithmetic operations, sequences, or even more complex relationships. Practicing with various types of number series and pairs helps in quickly identifying potential rules.

• Types of Patterns: Patterns can involve addition, subtraction, multiplication, division, squares, cubes, prime numbers, Fibonacci sequence, or a combination of operations.
• Strategy: Start by looking for simple relationships (like differences, ratios). If simple methods don't work, try combinations of operations or relationships based on the position of the number.