If `p` be a prime, ` p > 3` and let `x` be the product of positive integers `1,2,3….., (p - 1)` , then consider the following statements:
1. `x` is a composite number divisible by `p`.
2. `x` is a composite number not divisible by `p`, but some prime numbers greater than `p` may divide `x`.
3. `x` is not divisible by any prime `(p - 2)`.
4. All primes less than `(p -1)` divide `x`.
Of these statements :

Options

A. 1 and 2 are correct
B. 2 and 3 are correct
C. 3 and 4 are correct
D. 4 alone is correct

Answer

To solve the problem, we need to analyze each statement regarding the product \( x \) of the integers from \( 1 \) to \( p-1 \), where \( p \) is a prime number greater than \( 3 \).

### Step-by-Step Solution:

1. **Understanding \( x \)**:
- \( x = 1 \times 2 \times 3 \times \ldots \times (p - 1) = (p - 1)! \) (the factorial of \( p - 1 \)).
- Since \( p \) is a prime number greater than \( 3 \), \( p - 1 \) is an even number, and thus \( (p - 1)! \) is a composite number.

**Hint**: Remember that the factorial of any integer greater than 1 is always composite.

2. **Analyzing Statement 1**:
- Statement 1 claims that \( x \) is a composite number divisible by \( p \).
- Since \( p \) is prime and greater than \( 3 \), \( (p - 1)! \) contains all integers from \( 1 \) to \( p - 1 \), none of which are divisible by \( p \). Therefore, \( (p - 1)! \) is not divisible by \( p \).

**Hint**: A prime number cannot divide the factorial of numbers less than itself.

3. **Analyzing Statement 2**:
- Statement 2 claims that \( x \) is a composite number not divisible by \( p \), but some prime numbers greater than \( p \) may divide \( x \).
- While \( x \) is indeed composite and not divisible by \( p \), no prime numbers greater than \( p \) can divide \( (p - 1)! \) since they are larger than any of the factors in the factorial.

**Hint**: Consider the definition of divisibility and the range of numbers involved.

4. **Analyzing Statement 3**:
- Statement 3 claims that \( x \) is not divisible by any prime \( (p - 2) \).
- For example, if \( p = 5 \), then \( p - 2 = 3 \), and \( 24 \) (which is \( 4! \)) is divisible by \( 3 \). Thus, this statement is false.

**Hint**: Check specific examples to validate divisibility claims.

5. **Analyzing Statement 4**:
- Statement 4 claims that all primes less than \( (p - 1) \) divide \( x \).
- For \( p = 5 \), the primes less than \( 4 \) are \( 2 \) and \( 3 \). Both \( 2 \) and \( 3 \) divide \( 24 \), so this statement is true.

**Hint**: Verify the divisibility of smaller primes against the factorial.

### Conclusion:
- Statement 1 is false.
- Statement 2 is false.
- Statement 3 is false.
- Statement 4 is true.

Thus, the only correct statement is statement 4.

### Final Answer:
Only statement 4 is correct.