Let P (n) be the statement. “`P (n) = 16 n + 3` is prime. Is P (3) true ?

Answer

To determine whether \( P(3) \) is true, we need to evaluate the expression given in the statement \( P(n) = 16n + 3 \) for \( n = 3 \) and check if the result is a prime number.

### Step-by-Step Solution:

1. **Substitute \( n \) with 3 in the expression:**
\[ P(3) = 16(3) + 3 \]

2. **Calculate \( 16 \times 3 \):**
\[ 16 \times 3 = 48 \]

3. **Add 3 to the result:**
\[ P(3) = 48 + 3 = 51 \]

4. **Check if 51 is a prime number:**
- A prime number is defined as a number greater than 1 that has no positive divisors other than 1 and itself.
- The number 51 can be factored as \( 3 \times 17 \).

5. **Conclusion:**
Since 51 has divisors other than 1 and itself (specifically, 3 and 17), it is not a prime number. Therefore, \( P(3) \) is not true.

### Final Answer:
\( P(3) \) is not true because 51 is not a prime number.

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