Logic Identification in Number Pairs

The question asks us to identify a logical pattern between pairs of numbers and apply it to a new number. We are given two examples that follow the same rule:

• 9 is related to 16.
• 25 is related to 36.

Let's examine the first relationship: 9 and 16. We can see that both numbers are perfect squares:

• $9 = 3^2$
• $16 = 4^2$

The base numbers involved are 3 and 4. A clear connection is that 4 is one greater than 3 ($4 = 3 + 1$).

This suggests a pattern: If the first number is the square of an integer n (represented as $n^2$), the second number is the square of the next integer ($n+1$), represented as $(n+1)^2$.

To confirm this logic, let's test it with the second pair: 25 and 36.

• $25 = 5^2$
• $36 = 6^2$

Here, the base numbers are 5 and 6. Indeed, 6 is one greater than 5 ($6 = 5 + 1$). If we apply the pattern identified:

Starting with $25 = 5^2$, the related number should be $(5+1)^2 = 6^2 = 36$. This matches the given information, validating our logic.

Applying the Pattern to 81

Now, we need to find the number related to 81 using the same logic. First, express 81 as a perfect square:

• $81 = 9^2$

In this case, the base number is $n=9$. According to the pattern, the related number will be the square of ($n+1$):

• Related Number $= (9+1)^2$
• Related Number $= 10^2$
• Related Number $= 100$

Therefore, following the established number relation logic, 81 is related to 100.

Conclusion and Option Verification

The identified logic is that a number $n^2$ is related to $(n+1)^2$. Applying this to $81 = 9^2$, we find the related number is $(9+1)^2 = 10^2 = 100$.

Comparing this result with the given options:

The calculated value of 100 correctly corresponds to Option 2.