If `p = N + 5` when N is the product of any three consecutive postivie integer. Then :

Options

A. pis prime
B. p is odd
C. p is divisible by 6
D. either of (b), (c)

Answer

To solve the problem, we need to analyze the expression given and evaluate the options based on the value of \( p \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
We are given that \( p = N + 5 \) where \( N \) is the product of any three consecutive positive integers. Let's denote these three consecutive integers as \( n, n+1, n+2 \).

2. **Finding the Product**:
The product of three consecutive integers can be expressed as:
\[ N = n(n + 1)(n + 2) \]

3. **Calculating \( p \)**:
Substituting \( N \) into the equation for \( p \):
\[ p = n(n + 1)(n + 2) + 5 \]

4. **Testing with Examples**:
Let's take a few examples to see how \( p \) behaves.

- **Example 1**: Let \( n = 4 \):
\[ N = 4 \times 5 \times 6 = 120 \]
\[ p = 120 + 5 = 125 \]
- Check if \( p \) is prime: 125 is not prime (it is \( 5^3 \)).
- Check if \( p \) is odd: 125 is odd.
- Check if \( p \) is divisible by 6: 125 is not divisible by 6.

- **Example 2**: Let \( n = 9 \):
\[ N = 9 \times 10 \times 11 = 990 \]
\[ p = 990 + 5 = 995 \]
- Check if \( p \) is prime: 995 is not prime (it is \( 5 \times 199 \)).
- Check if \( p \) is odd: 995 is odd.
- Check if \( p \) is divisible by 6: 995 is not divisible by 6.

5. **Analyzing the Options**:
Based on the examples:
- \( p \) is not prime.
- \( p \) is odd.
- \( p \) is not divisible by 6.

Since both examples consistently show that \( p \) is odd, we can conclude that the correct option is that \( p \) is odd.

### Conclusion:
The correct answer is that \( p \) is odd.