Finding the Odd Number Pair: Analyzing Patterns

The question asks us to identify the pair of numbers that does not follow the same pattern as the other three pairs. We are given four pairs of numbers:

• 13 : 2186
• 7 : 336
• 11 : 1320
• 5 : 120

We need to examine each pair to find a consistent relationship between the first number and the second number. Let's look for common mathematical operations like squaring, cubing, multiplication, addition, or subtraction.

Identifying the Number Pattern

Let the first number in each pair be \(n\). We need to find a rule that transforms \(n\) into the second number for most of the pairs.

Let's test a common pattern involving cubes: \(n^3\).

• For the pair 5 : 120: \(5^3 = 5 \times 5 \times 5 = 125\). The second number is 120. The difference is \(125 - 120 = 5\). This suggests a pattern like \(n^3 - n\). Let's check if \(5^3 - 5\) equals 120. \(125 - 5 = 120\). Yes, it matches.
• For the pair 7 : 336: Let's test the pattern \(n^3 - n\). \(7^3 = 7 \times 7 \times 7 = 49 \times 7 = 343\). \(n^3 - n = 343 - 7 = 336\). Yes, it matches the second number in this pair.
• For the pair 11 : 1320: Let's test the pattern \(n^3 - n\). \(11^3 = 11 \times 11 \times 11 = 121 \times 11 = 1331\). \(n^3 - n = 1331 - 11 = 1320\). Yes, it matches the second number in this pair.

It appears that three of the given pairs follow the pattern where the second number is the cube of the first number minus the first number itself (\(n^3 - n\)).

Analyzing the Odd Pair

Now let's examine the first pair, 13 : 2186, using the identified pattern \(n^3 - n\):

• For the pair 13 : 2186: Let \(n = 13\). We calculate \(13^3 - 13\). \(13^3 = 13 \times 13 \times 13 = 169 \times 13 = 2197\). \(n^3 - n = 2197 - 13 = 2184\).

The calculated value based on the pattern (\(2184\)) is not equal to the given second number in the pair (\(2186\)). Therefore, the pair 13 : 2186 does not follow the pattern \(n : n^3 - n\).

Since options 2, 3, and 4 follow the same pattern, the pair that is different or the odd one out is 13 : 2186.

Summary of Patterns

Let's summarize the relationship for each pair:

• 13 : 2186 — \(13^3 - 13 = 2197 - 13 = 2184 \neq 2186\). This pair does NOT follow the pattern.
• 7 : 336 — \(7^3 - 7 = 343 - 7 = 336\). This pair follows the pattern.
• 11 : 1320 — \(11^3 - 11 = 1331 - 11 = 1320\). This pair follows the pattern.
• 5 : 120 — \(5^3 - 5 = 125 - 5 = 120\). This pair follows the pattern.

Based on this analysis, the pair 13 : 2186 is the one that is different from the others.

Conclusion

The pair of numbers 13 : 2186 is the odd one out because it does not fit the pattern \(n : n^3 - n\) that the other three pairs (7 : 336, 11 : 1320, and 5 : 120) follow.

Revision Table: Number Pattern Analysis

Reviewing the process helps solidify the understanding of finding patterns in number series.

• Identify the type of relationship: The second number is much larger, suggesting powers.
• Test common power-related patterns: \(n^2\), \(n^3\), \(n^2 \pm c\), \(n^3 \pm c\), \(n^3 \pm n\), etc.
• Apply the potential pattern to at least three options to check consistency.
• Verify if the remaining option follows the pattern. If not, it's likely the odd one out.

Additional Information: Types of Number Series Patterns

Numerical reasoning questions often involve various patterns. Besides the \(n^3 - n\) pattern seen here, other common types include:

• Arithmetic Progression: Adding or subtracting a constant value.
• Geometric Progression: Multiplying or dividing by a constant value.
• Difference Series: The difference between consecutive terms follows a pattern.
• Mixed Operations: Combining addition, subtraction, multiplication, division, powers, etc.
• Alternating Patterns: Different rules apply to alternate terms.
• Digit Based Patterns: Patterns based on the digits of the numbers.
• Cube/Square Related: Patterns involving \(n^2\), \(n^3\), or operations on them.

Practicing different types of number pattern problems helps improve pattern recognition skills.