If `n^2 = 123454321`, then the value of n is :

Options

A. 1001
B. not a natural number
C. 111111
D. none of these

Answer

To find the value of \( n \) given that \( n^2 = 123454321 \), we need to calculate the square root of \( 123454321 \).

### Step-by-Step Solution:

1. **Identify the equation**: We start with the equation \( n^2 = 123454321 \).

2. **Take the square root**: To find \( n \), we need to take the square root of both sides of the equation:
\[ n = \sqrt{123454321} \]

3. **Estimate the square root**: We can estimate the square root by recognizing that \( 123454321 \) is a large number. We can look for perfect squares around this number.

4. **Check for perfect squares**: We can check if \( n \) is a whole number by calculating:
- \( 111^2 = 12321 \)
- \( 1111^2 = 1234321 \)
- \( 11111^2 = 12345654321 \)

From the above calculations, we see that \( 11111^2 \) is too large, and \( 1111^2 \) is too small.

5. **Calculate directly**: Now, we will calculate \( \sqrt{123454321} \) directly. Using a calculator or long division method, we find:
\[ n = 11111 \]

6. **Verify the result**: To ensure our answer is correct, we can square \( 11111 \) to check:
\[ 11111^2 = 123454321 \]

Thus, the value of \( n \) is \( 11111 \).

### Final Answer:
\[ n = 11111 \]