Finding the Number Relationship Pattern

The question asks us to identify the set of numbers that shares the same mathematical relationship as the given sets: (14, 10, 296) and (8, 17, 353). We need to find a rule that connects the first two numbers to the third number in each set, using operations on the whole numbers themselves, not their individual digits.

Analyzing the Given Number Sets

Let's look at the first set: (14, 10, 296). Let the numbers be \(A=14\), \(B=10\), and \(C=296\). We need to find a relationship between 14, 10, and 296.

• Adding the first two numbers: \(14 + 10 = 24\). This is not 296.
• Multiplying the first two numbers: \(14 \times 10 = 140\). This is not 296.
• Squaring the first number: \(14^2 = 196\).
• Squaring the second number: \(10^2 = 100\).
• Adding the squares of the first two numbers: \(14^2 + 10^2 = 196 + 100 = 296\). This matches the third number in the set.

Let's test this potential pattern (\(A^2 + B^2 = C\)) with the second given set: (8, 17, 353). Let \(A=8\), \(B=17\), and \(C=353\).

• Squaring the first number: \(8^2 = 64\).
• Squaring the second number: \(17^2 = 289\).
• Adding the squares of the first two numbers: \(8^2 + 17^2 = 64 + 289 = 353\). This also matches the third number in the set.

The pattern seems to be that the third number in the set is the sum of the squares of the first two numbers (\(C = A^2 + B^2\)).

Checking the Options Based on the Pattern

Now we will apply the identified pattern (\(A^2 + B^2 = C\)) to each of the given options to see which set follows the same rule.

Option 1: (29, 7, 690)

• Let \(A=29\), \(B=7\).
• Calculate \(A^2 + B^2\): \(29^2 + 7^2 = 841 + 49 = 890\).
• The third number in the set is 690. Since \(890 \neq 690\), this set does not follow the pattern.

Option 2: (18, 6, 576)

• Let \(A=18\), \(B=6\).
• Calculate \(A^2 + B^2\): \(18^2 + 6^2 = 324 + 36 = 360\).
• The third number in the set is 576. Since \(360 \neq 576\), this set does not follow the pattern.

Option 3: (5, 13, 194)

• Let \(A=5\), \(B=13\).
• Calculate \(A^2 + B^2\): \(5^2 + 13^2 = 25 + 169 = 194\).
• The third number in the set is 194. Since \(194 = 194\), this set follows the pattern.

Option 4: (12, 3, 135)

• Let \(A=12\), \(B=3\).
• Calculate \(A^2 + B^2\): \(12^2 + 3^2 = 144 + 9 = 153\).
• The third number in the set is 135. Since \(153 \neq 135\), this set does not follow the pattern.

Only the set (5, 13, 194) follows the established pattern where the third number is the sum of the squares of the first two numbers.

Conclusion

By analyzing the relationship in the given sets, we found the pattern where the third number is the sum of the squares of the first two numbers. Applying this pattern to the options revealed that the set (5, 13, 194) is the only one that follows the same relationship.

Revision Table: Number Pattern Analysis

Understanding number patterns is crucial for logical reasoning questions. Here's a quick summary of steps:

• Examine the relationship between the numbers in the given sets.
• Test common mathematical operations (addition, subtraction, multiplication, division, squaring, cubing, etc.) and combinations thereof.
• Once a pattern is found that works for all given sets, apply it to the options.
• Verify which option correctly follows the established pattern.

Additional Information: Solving Number Series and Analogies

Questions involving number sets and patterns are often seen in logical reasoning and quantitative aptitude sections of competitive exams. They test your ability to quickly identify underlying mathematical rules or sequences. Common patterns include arithmetic progressions, geometric progressions, series based on squares or cubes, alternating patterns, and relationships between numbers within a set using basic operations or combinations of operations.

Practicing with different types of number patterns helps in developing the skill to recognize them faster during exams. It's important to test various simple operations first before looking for more complex relationships.