Finding the Relationship in Number Sets

The question asks us to identify the option where the set of numbers shares the same mathematical relationship as the numbers in the two given sets. We need to analyze the pattern within the provided sets first.

Analyzing the Given Number Sets

Let's examine the first set: (11, 121, 1331).

• The first number is 11.
• The second number is 121. We notice that $11 \times 11 = 121$. This means the second number is the square of the first number ($\text{11}^2$).
• The third number is 1331. We can check if there is a relationship involving the first number. $11 \times 11 \times 11 = 121 \times 11 = 1331$. This means the third number is the cube of the first number ($\text{11}^3$).

So, the relationship in the first set is $(\text{A}, \text{A}^2, \text{A}^3)$, where A is the first number.

Let's examine the second set: (13, 169, 2197).

• The first number is 13.
• The second number is 169. We check the square: $13 \times 13 = 169$. This matches the second number ($\text{13}^2$).
• The third number is 2197. We check the cube: $13 \times 13 \times 13 = 169 \times 13 = 2197$. This matches the third number ($\text{13}^3$).

The second set also follows the relationship $(\text{A}, \text{A}^2, \text{A}^3)$, where A is the first number.

The consistent relationship across both given sets is that the numbers follow the pattern (First number, First number squared, First number cubed).

Mathematically, this is $(\text{A}, \text{A}^2, \text{A}^3)$.

Evaluating the Options Based on the Relationship (A, A², A³)

Now, we will test each option to see which one fits the pattern $(\text{A}, \text{A}^2, \text{A}^3)$.

Option 1: (9, 729, 81)

• First number (A) = 9
• Second number: $\text{A}^2 = 9^2 = 9 \times 9 = 81$. The given second number is 729. This does not match.
• Third number: $\text{A}^3 = 9^3 = 9 \times 9 \times 9 = 81 \times 9 = 729$. The given third number is 81. This does not match.

This option does not follow the $(\text{A}, \text{A}^2, \text{A}^3)$ relationship.

Option 2: (6, 36, 324)

• First number (A) = 6
• Second number: $\text{A}^2 = 6^2 = 6 \times 6 = 36$. The given second number is 36. This matches.
• Third number: $\text{A}^3 = 6^3 = 6 \times 6 \times 6 = 36 \times 6 = 216$. The given third number is 324. This does not match.

This option does not follow the $(\text{A}, \text{A}^2, \text{A}^3)$ relationship.

Option 3: (5, 25, 625)

• First number (A) = 5
• Second number: $\text{A}^2 = 5^2 = 5 \times 5 = 25$. The given second number is 25. This matches.
• Third number: $\text{A}^3 = 5^3 = 5 \times 5 \times 5 = 25 \times 5 = 125$. The given third number is 625. This does not match. (Note: 625 is actually $5^4$).

This option does not follow the $(\text{A}, \text{A}^2, \text{A}^3)$ relationship.

Option 4: (7, 49, 343)

• First number (A) = 7
• Second number: $\text{A}^2 = 7^2 = 7 \times 7 = 49$. The given second number is 49. This matches.
• Third number: $\text{A}^3 = 7^3 = 7 \times 7 \times 7 = 49 \times 7 = 343$. The given third number is 343. This matches.

This option perfectly follows the $(\text{A}, \text{A}^2, \text{A}^3)$ relationship.

Conclusion

Based on our analysis, the set (7, 49, 343) is the only option that shares the same $(\text{A}, \text{A}^2, \text{A}^3)$ relationship as the given sets (11, 121, 1331) and (13, 169, 2197).

Revision Table: Checking the Relationship (A, A², A³)

Additional Information: Number Analogy and Reasoning

Number analogy questions test your ability to find patterns and relationships between numbers. These relationships can involve basic arithmetic operations (addition, subtraction, multiplication, division), squares, cubes, prime numbers, digits, or combinations of these. The key is to analyze the given examples carefully to deduce the rule before applying it to the options.

In this type of question, always perform operations on the whole numbers as instructed, unless the question specifically mentions breaking down digits.

Common patterns include:

• (A, A+k, A+2k) or (A, A*k, A*k²)
• (A, A², A³)
• (A, A², A³ + k)
• (A, A+k, B, B+k)
• Relationships based on prime numbers, composite numbers, etc.

Practicing different types of number series and analogy problems helps in quickly identifying potential relationships.