Consider the statement `P(n): n^2 - n + 41` is prime. Them which of the following is true?

Options

A. Both P(3) and P(5) are true
B. P(3) is true but P(5) is false
C. Both P(3) and P(5) are false.
D. P(3) is false but P(5) is true.

Answer

To solve the problem, we need to evaluate the statement \( P(n): n^2 - n + 41 \) for \( n = 3 \) and \( n = 5 \) to determine whether the outputs are prime numbers.

### Step 1: Calculate \( P(3) \)
We start by substituting \( n = 3 \) into the expression:

\[ P(3) = 3^2 - 3 + 41 \]

Calculating \( 3^2 \):

\[ 3^2 = 9 \]

Now substituting back into the equation:

\[ P(3) = 9 - 3 + 41 \]

Calculating \( 9 - 3 \):

\[ 9 - 3 = 6 \]

Now adding 41:

\[ P(3) = 6 + 41 = 47 \]

Now we check if 47 is a prime number. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. Since 47 can only be divided by 1 and 47, it is indeed a prime number.

### Conclusion for \( P(3) \):
Thus, \( P(3) \) is true because 47 is prime.

### Step 2: Calculate \( P(5) \)
Next, we substitute \( n = 5 \) into the expression:

\[ P(5) = 5^2 - 5 + 41 \]

Calculating \( 5^2 \):

\[ 5^2 = 25 \]

Now substituting back into the equation:

\[ P(5) = 25 - 5 + 41 \]

Calculating \( 25 - 5 \):

\[ 25 - 5 = 20 \]

Now adding 41:

\[ P(5) = 20 + 41 = 61 \]

Now we check if 61 is a prime number. Since 61 can only be divided by 1 and 61, it is also a prime number.

### Conclusion for \( P(5) \):
Thus, \( P(5) \) is true because 61 is prime.

### Final Conclusion:
Both \( P(3) \) and \( P(5) \) are true. Therefore, the correct option is:

**Option A: Both \( P(3) \) and \( P(5) \) are true.**

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