Finding the Number Pair That Doesn't Belong

The question asks us to identify the number pair that does not fit the pattern followed by the other pairs in the given group: (1, 1), (4, 64), (8, 512), and (9, 719). To solve this, we need to look for a common relationship or rule that connects the numbers within each pair.

Analyzing the Number Pairs and Potential Patterns

Let's examine each pair and see if we can find a mathematical operation or relationship between the first number (let's call it $x$) and the second number (let's call it $y$).

• Pair 1: (1, 1)
  Here, $x=1$ and $y=1$. Possible relationships: $1^1=1$, $1^2=1$, $1^3=1$, etc.
• Pair 2: (4, 64)
  Here, $x=4$ and $y=64$. Let's test simple relationships:
  Addition: $4 + a = 64 \implies a = 60$. This is unlikely to be a general pattern.
  Multiplication: $4 \times a = 64 \implies a = 16$. Also unlikely to be a general pattern.
  Powers: $4^2 = 16$, $4^3 = 4 \times 4 \times 4 = 16 \times 4 = 64$. This fits! It seems $y = x^3$.
• Pair 3: (8, 512)
  Here, $x=8$ and $y=512$. Let's test the potential pattern $y = x^3$:
  $8^3 = 8 \times 8 \times 8 = 64 \times 8 = 512$. This also fits the pattern $y = x^3$.

Testing the Pattern on All Pairs

The pattern $y = x^3$ seems to hold for the pairs (4, 64) and (8, 512). Let's verify this pattern for the first pair (1, 1):

• Pair 1: (1, 1)
  Is $1^3 = 1$? Yes, $1 \times 1 \times 1 = 1$. This pair also fits the pattern $y = x^3$.

Now let's test the last pair (9, 719) against the discovered pattern $y = x^3$:

• Pair 4: (9, 719)
  Here, $x=9$ and $y=719$. Is $9^3 = 719$?
  $9^3 = 9 \times 9 \times 9 = 81 \times 9 = 729$.
  The calculated value is 729, but the given second number is 719. So, this pair does NOT follow the pattern $y = x^3$.

Identifying the Odd Pair Out

Based on our analysis, three of the number pairs follow the pattern where the second number is the cube of the first number:

• (1, 1) because $1^3 = 1$
• (4, 64) because $4^3 = 64$
• (8, 512) because $8^3 = 512$

The pair (9, 719) does not follow this pattern because $9^3 = 729$, not 719.

Therefore, the number pair that does NOT belong in the group is (9, 719).

Revision Table - Cube Patterns

Understanding perfect cubes is useful for solving pattern-based questions like this one.

Additional Information - Logical Reasoning Patterns

Logical reasoning questions often involve identifying patterns in numbers or sequences. Common types of patterns include:

• Arithmetic progression (constant difference)
• Geometric progression (constant ratio)
• Square numbers ($1^2, 2^2, 3^2, \dots$)
• Cube numbers ($1^3, 2^3, 3^3, \dots$)
• Combinations of operations (e.g., $2x+1$, $x^2-1$)
• Fibonacci sequence (sum of previous two numbers)
• Alternating patterns
• Digit-based operations

To solve these questions, it's helpful to test simple patterns first and look for a rule that applies consistently to most elements in the series or group. The element that deviates from this rule is usually the odd one out.