If the numbers `x, 2 x + 3` are the primes, then x is equal to

Options

A. 3
B. 2
C. 11
D. 13

Answer

To solve the problem, we need to determine the value of \( x \) such that both \( x \) and \( 2x + 3 \) are prime numbers.

### Step-by-step Solution:

1. **Understanding the Problem**: We need to find values of \( x \) such that both \( x \) and \( 2x + 3 \) are prime numbers.

2. **Testing Possible Values of \( x \)**: We will test different integer values for \( x \) to see if both numbers are prime.

3. **Testing \( x = 1 \)**:
- Calculate \( 2x + 3 \):
\[ 2(1) + 3 = 2 + 3 = 5 \]
- Check if both \( 1 \) and \( 5 \) are prime:
- \( 1 \) is not a prime number.
- Conclusion: This value does not work.

4. **Testing \( x = 2 \)**:
- Calculate \( 2x + 3 \):
\[ 2(2) + 3 = 4 + 3 = 7 \]
- Check if both \( 2 \) and \( 7 \) are prime:
- \( 2 \) is a prime number.
- \( 7 \) is a prime number.
- Conclusion: This value works.

5. **Testing \( x = 3 \)**:
- Calculate \( 2x + 3 \):
\[ 2(3) + 3 = 6 + 3 = 9 \]
- Check if both \( 3 \) and \( 9 \) are prime:
- \( 3 \) is a prime number.
- \( 9 \) is not a prime number.
- Conclusion: This value does not work.

6. **Testing \( x = 11 \)**:
- Calculate \( 2x + 3 \):
\[ 2(11) + 3 = 22 + 3 = 25 \]
- Check if both \( 11 \) and \( 25 \) are prime:
- \( 11 \) is a prime number.
- \( 25 \) is not a prime number.
- Conclusion: This value does not work.

7. **Testing \( x = 13 \)**:
- Calculate \( 2x + 3 \):
\[ 2(13) + 3 = 26 + 3 = 29 \]
- Check if both \( 13 \) and \( 29 \) are prime:
- \( 13 \) is a prime number.
- \( 29 \) is a prime number.
- Conclusion: This value works.

8. **Final Conclusion**: The values of \( x \) that satisfy the condition are \( 2 \) and \( 13 \).

Thus, \( x \) can be either \( 2 \) or \( 13 \).