If `P(n)` be the statement "`10n+3` is a prime number", then prove that P(1) and P(2) are true but P(3) is false.

Answer

To solve the problem, we need to evaluate the statements \( P(1) \), \( P(2) \), and \( P(3) \) based on the definition of \( P(n) \): that \( 10n + 3 \) is a prime number.

### Step 1: Evaluate \( P(1) \)

We start with \( P(1) \):
\[ P(1) = 10 \cdot 1 + 3 = 10 + 3 = 13 \]
Now we check if 13 is a prime number. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The only divisors of 13 are 1 and 13.

Thus, \( P(1) \) is true because 13 is a prime number.

### Step 2: Evaluate \( P(2) \)

Next, we evaluate \( P(2) \):
\[ P(2) = 10 \cdot 2 + 3 = 20 + 3 = 23 \]
Now we check if 23 is a prime number. The only divisors of 23 are 1 and 23.

Thus, \( P(2) \) is true because 23 is a prime number.

### Step 3: Evaluate \( P(3) \)

Now, we evaluate \( P(3) \):
\[ P(3) = 10 \cdot 3 + 3 = 30 + 3 = 33 \]
Now we check if 33 is a prime number. The divisors of 33 are 1, 3, 11, and 33.

Thus, \( P(3) \) is false because 33 is not a prime number (it has divisors other than 1 and itself).

### Conclusion

We have shown that:
- \( P(1) \) is true (13 is prime).
- \( P(2) \) is true (23 is prime).
- \( P(3) \) is false (33 is not prime).

Therefore, we conclude that \( P(1) \) and \( P(2) \) are true, but \( P(3) \) is false.

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