If `{p in N: "p is a prime" and p=(7n^(2)+3n+3)/(n)"for some n" in N}'` then the number of elements in the set A, is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to find the number of elements in the set \( A = \{ p \in \mathbb{N} : p \text{ is a prime and } p = \frac{7n^2 + 3n + 3}{n} \text{ for some } n \in \mathbb{N} \} \).

### Step-by-step Solution:

1. **Understanding the Expression for \( p \)**:
We start with the expression given for \( p \):
\[ p = \frac{7n^2 + 3n + 3}{n} \]
We can simplify this expression:
\[ p = 7n + 3 + \frac{3}{n} \]

2. **Conditions for \( p \)**:
We need \( p \) to be a natural number. This means \( \frac{3}{n} \) must also be a natural number, which implies that \( n \) must be a divisor of 3. The natural divisors of 3 are \( n = 1 \) and \( n = 3 \).

3. **Calculating \( p \) for Valid \( n \)**:
- **For \( n = 1 \)**:
\[ p = 7(1) + 3 + \frac{3}{1} = 7 + 3 + 3 = 13 \]
Now, we check if 13 is a prime number. Yes, 13 is a prime number.

- **For \( n = 3 \)**:
\[ p = 7(3) + 3 + \frac{3}{3} = 21 + 3 + 1 = 25 \]
Now, we check if 25 is a prime number. No, 25 is not a prime number (it is \( 5 \times 5 \)).

4. **Identifying Elements in Set \( A \)**:
From the calculations above, the only value of \( p \) that is a prime number is when \( n = 1 \), which gives us \( p = 13 \).

5. **Conclusion**:
Therefore, the set \( A \) contains only one element, which is \( 13 \).

Thus, the number of elements in the set \( A \) is:
\[ \text{Number of elements in set } A = 1 \]

### Final Answer:
The number of elements in the set \( A \) is \( 1 \).