If `n = 1 +x` and x is the product of four consecutive integers , then which of the following is true ?
I n is an odd integer .
II . N is prime
III. N is a perfect square .

Options

A. Only I is correct
B. Only III is correct
C. Both I and II are correct
D. Both I and III are correct

Answer

To solve the problem, we need to analyze the statements given about \( n \), where \( n = 1 + x \) and \( x \) is the product of four consecutive integers.

### Step-by-Step Solution:

1. **Define the Product of Four Consecutive Integers**:
Let the four consecutive integers be \( k, k+1, k+2, k+3 \). Therefore, the product \( x \) can be expressed as:
\[ x = k(k+1)(k+2)(k+3) \]

2. **Calculate \( n \)**:
Now, substituting \( x \) into the equation for \( n \):
\[ n = 1 + x = 1 + k(k+1)(k+2)(k+3) \]

3. **Check if \( n \) is an Odd Integer**:
The product \( k(k+1)(k+2)(k+3) \) consists of four consecutive integers. Among any four consecutive integers, at least one will be even, making the product \( x \) even. Therefore:
\[ n = 1 + \text{(even number)} = \text{odd number} \]
Thus, **Statement I** is true.

4. **Check if \( n \) is Prime**:
We need to check if \( n \) can be a prime number. For example:
- If \( k = 1 \): \( x = 1 \cdot 2 \cdot 3 \cdot 4 = 24 \) → \( n = 1 + 24 = 25 \) (not prime)
- If \( k = 2 \): \( x = 2 \cdot 3 \cdot 4 \cdot 5 = 120 \) → \( n = 1 + 120 = 121 \) (not prime)
- If \( k = 3 \): \( x = 3 \cdot 4 \cdot 5 \cdot 6 = 360 \) → \( n = 1 + 360 = 361 \) (not prime)

It appears that \( n \) is not prime for these examples. Thus, **Statement II** is false.

5. **Check if \( n \) is a Perfect Square**:
Continuing from the previous examples:
- For \( k = 1 \): \( n = 25 \) (which is \( 5^2 \), a perfect square)
- For \( k = 2 \): \( n = 121 \) (which is \( 11^2 \), a perfect square)
- For \( k = 3 \): \( n = 361 \) (which is \( 19^2 \), a perfect square)

In all these cases, \( n \) is a perfect square. Thus, **Statement III** is true.

### Conclusion:
- **Statement I**: True (n is an odd integer)
- **Statement II**: False (n is not prime)
- **Statement III**: True (n is a perfect square)

The correct statements are I and III.