Which one of the following is true?
(i) The least positive value of `n` for which `n!` can be divided by `(n + 1)` is `5`
(ii) The square root of `123454321` is `1111`
(iii) The sum of the squares of the first `24` natural numbers is a perfect square

Options

A. (i) and (ii)
B. (ii) and (iii)
C. (i) and (iii)
D. all of these

Answer

To determine which of the three statements is true, we will analyze each statement one by one.

### Statement (i): The least positive value of `n` for which `n!` can be divided by `(n + 1)` is `5`.

1. **Understanding Factorials**: The factorial of a number `n`, denoted as `n!`, is the product of all positive integers up to `n`. We need to check if `n!` is divisible by `n + 1`.

2. **Testing Values of `n`**:
- For `n = 1`:
\[ 1! = 1 \quad \text{and} \quad 1 + 1 = 2 \quad \Rightarrow \quad 1 \text{ is not divisible by } 2 \]
- For `n = 2`:
\[ 2! = 2 \quad \text{and} \quad 2 + 1 = 3 \quad \Rightarrow \quad 2 \text{ is not divisible by } 3 \]
- For `n = 3`:
\[ 3! = 6 \quad \text{and} \quad 3 + 1 = 4 \quad \Rightarrow \quad 6 \text{ is not divisible by } 4 \]
- For `n = 4`:
\[ 4! = 24 \quad \text{and} \quad 4 + 1 = 5 \quad \Rightarrow \quad 24 \text{ is not divisible by } 5 \]
- For `n = 5`:
\[ 5! = 120 \quad \text{and} \quad 5 + 1 = 6 \quad \Rightarrow \quad 120 \text{ is divisible by } 6 \]

3. **Conclusion for Statement (i)**: The least positive value of `n` for which `n!` can be divided by `(n + 1)` is indeed `5`. Therefore, this statement is **true**.

### Statement (ii): The square root of `123454321` is `1111`.

1. **Calculating the Square Root**:
- To verify this, we can calculate \( 1111^2 \):
\[ 1111 \times 1111 = 1234321 \]
- Since \( 1234321 \) is not equal to \( 123454321 \), this statement is **false**.

### Statement (iii): The sum of the squares of the first `24` natural numbers is a perfect square.

1. **Using the Formula for the Sum of Squares**:
- The formula for the sum of the squares of the first `n` natural numbers is:
\[ S = \frac{n(n + 1)(2n + 1)}{6} \]
- For \( n = 24 \):
\[ S = \frac{24 \times 25 \times 49}{6} \]
- Simplifying:
\[ S = \frac{24 \times 25 \times 49}{6} = 4 \times 25 \times 49 = 4900 \]
- Now, we check if \( 4900 \) is a perfect square:
\[ \sqrt{4900} = 70 \]
- Since \( 70^2 = 4900 \), this statement is **true**.

### Final Conclusion:
- Statement (i) is true.
- Statement (ii) is false.
- Statement (iii) is true.

Thus, the true statements are (i) and (iii).