Finding the Number Relationship in Sets

The question asks us to identify the relationship between the numbers in the given sets and then find an option set that shares the same relationship. We are given two sets: (7, 559, 6) and (8, 637, 5).

The rule specifies that operations must be performed on the whole numbers provided, not on their constituent digits.

Analyzing the Given Sets to Find the Pattern

Let's examine the numbers in the first set (7, 559, 6).

• The first number is 7.
• The third number is 6.
• The middle number is 559.

We need to find a relationship between 7, 6, and 559 using mathematical operations. Let's consider basic operations and powers:

• Sum: 7 + 6 = 13 (Not 559)
• Difference: |7 - 6| = 1 (Not 559)
• Product: 7 × 6 = 42 (Not 559)
• Squares: $7^2 = 49$, $6^2 = 36$. Sum of squares: $49 + 36 = 85$ (Not 559)
• Cubes: $7^3 = 343$, $6^3 = 216$. Sum of cubes: $343 + 216 = 559$.

The sum of the cubes of the first and third numbers equals the middle number in the first set:

\(7^3 + 6^3 = 343 + 216 = 559\)

Let's check if this relationship holds for the second given set (8, 637, 5).

• The first number is 8.
• The third number is 5.
• The middle number is 637.

Using the discovered pattern, let's calculate the sum of the cubes of the first and third numbers:

\(8^3 + 5^3 = 512 + 125 = 637\)

This also matches the middle number in the second set.

So, the established relationship is: If a set is given as $(a, b, c)$, then $b = a^3 + c^3$.

Verifying the Options using the Relationship

Now we will check each option to see which one follows the pattern \(b = a^3 + c^3\), where $a$ is the first number, $b$ is the middle number, and $c$ is the third number.

Option 1: (9, 1241, 8)

• Here, $a=9$, $b=1241$, $c=8$.
• Let's calculate $a^3 + c^3$: \(9^3 + 8^3 = 729 + 512 = 1241\).
• The calculated value 1241 matches the middle number $b$.
• So, this option follows the relationship.

Option 2: (8, 755, 7)

• Here, $a=8$, $b=755$, $c=7$.
• Let's calculate $a^3 + c^3$: \(8^3 + 7^3 = 512 + 343 = 855\).
• The calculated value 855 does not match the middle number $b$ (755).
• So, this option does not follow the relationship.

Option 3: (10, 1729, 8)

• Here, $a=10$, $b=1729$, $c=8$.
• Let's calculate $a^3 + c^3$: \(10^3 + 8^3 = 1000 + 512 = 1512\).
• The calculated value 1512 does not match the middle number $b$ (1729).
• So, this option does not follow the relationship.

Option 4: (11, 1115, 6)

• Here, $a=11$, $b=1115$, $c=6$.
• Let's calculate $a^3 + c^3$: \(11^3 + 6^3 = 1331 + 216 = 1547\).
• The calculated value 1547 does not match the middle number $b$ (1115).
• So, this option does not follow the relationship.

Conclusion

Only Option 1 exhibits the same number relationship as the given sets, where the middle number is the sum of the cubes of the first and third numbers.

Revision Table: Key Concepts in Number Relationships

Additional Information: Cube Numbers and Reasoning

Cube numbers play a significant role in many reasoning problems involving number patterns. A cube number is the result of multiplying an integer by itself three times (i.e., $n \times n \times n$ or $n^3$).

Understanding the first few cube numbers can be helpful in solving such problems quickly:

• $1^3 = 1$
• $2^3 = 8$
• $3^3 = 27$
• $4^3 = 64$
• $5^3 = 125$
• $6^3 = 216$
• $7^3 = 343$
• $8^3 = 512$
• $9^3 = 729$
• $10^3 = 1000$
• $11^3 = 1331$
• $12^3 = 1728$

In problems like this, look for relationships involving sums, differences, products, or quotients of powers (squares, cubes) of the given numbers. Sometimes, relationships might involve the sum or product of digits, but the problem statement here explicitly forbids that.