If `N,(N+2)` and `(N+4)` are prime numbers, then the number of possible solutions for N are

Options

A. 1
B. 2
C. 3
D. None of these

Answer

To solve the problem, we need to determine the values of \( N \) such that \( N \), \( N+2 \), and \( N+4 \) are all prime numbers. Let's go through the steps systematically.

### Step 1: Understand the conditions
We need to find integers \( N \) such that:
- \( N \) is prime
- \( N + 2 \) is prime
- \( N + 4 \) is prime

### Step 2: Check small prime numbers
We will start checking small prime numbers for \( N \).

1. **If \( N = 2 \)**:
- \( N + 2 = 4 \) (not prime)
- So, \( N = 2 \) is not a solution.

2. **If \( N = 3 \)**:
- \( N + 2 = 5 \) (prime)
- \( N + 4 = 7 \) (prime)
- All three numbers (3, 5, and 7) are prime. So, \( N = 3 \) is a solution.

3. **If \( N = 5 \)**:
- \( N + 2 = 7 \) (prime)
- \( N + 4 = 9 \) (not prime)
- So, \( N = 5 \) is not a solution.

4. **If \( N = 7 \)**:
- \( N + 2 = 9 \) (not prime)
- So, \( N = 7 \) is not a solution.

5. **If \( N = 11 \)**:
- \( N + 2 = 13 \) (prime)
- \( N + 4 = 15 \) (not prime)
- So, \( N = 11 \) is not a solution.

6. **If \( N = 13 \)**:
- \( N + 2 = 15 \) (not prime)
- So, \( N = 13 \) is not a solution.

7. **If \( N = 17 \)**:
- \( N + 2 = 19 \) (prime)
- \( N + 4 = 21 \) (not prime)
- So, \( N = 17 \) is not a solution.

### Step 3: Analyze the results
From our checks, the only value of \( N \) that satisfies the condition where \( N \), \( N + 2 \), and \( N + 4 \) are all prime is \( N = 3 \).

### Conclusion
Thus, the number of possible solutions for \( N \) is **1**.