Consider the statement `P(n) : 10 n +3` is prime, then which of the following is not true?

Options

A. (a)`P(1)`
B. (b) `P(2)`
C. (c) `P(3)`
D. (d) `P(4)`

Answer

To solve the problem, we need to evaluate the statement \( P(n) : 10n + 3 \) for the values of \( n \) given in the options and determine which one is not a prime number.

### Step-by-Step Solution:

1. **Evaluate \( P(1) \)**:
\[ P(1) = 10 \times 1 + 3 = 10 + 3 = 13 \]
- **Check if 13 is prime**: The factors of 13 are 1 and 13. Therefore, 13 is a prime number.

2. **Evaluate \( P(2) \)**:
\[ P(2) = 10 \times 2 + 3 = 20 + 3 = 23 \]
- **Check if 23 is prime**: The factors of 23 are 1 and 23. Therefore, 23 is a prime number.

3. **Evaluate \( P(3) \)**:
\[ P(3) = 10 \times 3 + 3 = 30 + 3 = 33 \]
- **Check if 33 is prime**: The factors of 33 are 1, 3, 11, and 33. Therefore, 33 is not a prime number (it is composite).

4. **Evaluate \( P(4) \)**:
\[ P(4) = 10 \times 4 + 3 = 40 + 3 = 43 \]
- **Check if 43 is prime**: The factors of 43 are 1 and 43. Therefore, 43 is a prime number.

### Conclusion:
From the evaluations:
- \( P(1) = 13 \) (prime)
- \( P(2) = 23 \) (prime)
- \( P(3) = 33 \) (not prime, composite)
- \( P(4) = 43 \) (prime)

The option that is not true (i.e., not prime) is **option C: \( P(3) \)**.

### Final Answer:
**Option C: \( P(3) \)** is not true.

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