X10 is a base ten 3 digit prime number with the left most digits being 1 then 6. Y10 is a base ten 2 digit prime number
with the left most digit being 8. P2 and Q2 are both binary numbers given by P2 = X10 AND Y10 and Q2 = X10 OR Y10.
If P2 has a single 1 bit and Q2 has a single 0 bit, find X10 and Y10.

Let's solve this step-by-step. We need to find two prime numbers \( X_{10} \) and \( Y_{10} \) with specific properties and then determine two binary numbers \( P_2 \) and \( Q_2 \) based on the bitwise operations AND and OR.

### Problem Analysis

1. **\( X_{10} \)**: A three-digit prime number with its leftmost digits being 1 and 6, meaning it's of the form \( 16a \), where \( a \) is any digit from 0 to 9.
2. **\( Y_{10} \)**: A two-digit prime number with the leftmost digit being 8, meaning it is of the form \( 8b \), where \( b \) is a digit from 0 to 9.
3. **\( P_2 \) and \( Q_2 \)** are defined as:
   - \( P_2 = X_{10} \text{ AND } Y_{10} \)
   - \( Q_2 = X_{10} \text{ OR } Y_{10} \)
4. **Properties of \( P_2 \) and \( Q_2 \)**:
   - \( P_2 \) has a single 1 bit, which means \( P_2 \) is a power of 2 (e.g., 1, 2, 4, 8, etc.).
   - \( Q_2 \) has a single 0 bit, indicating that it is a binary number with one 0 bit among all 1s.

### Solution Approach

1. **Identify possible values of \( X_{10} \)**:
   - We need prime numbers of the form \( 16a \). Possible values are \( 163, 167, \) and \( 169 \). However, \( 169 \) is not prime, leaving \( 163 \) and \( 167 \).

2. **Identify possible values of \( Y_{10} \)**:
   - We need prime numbers of the form \( 8b \). Possible values are \( 83 \) and \( 89 \).

3. **Check the conditions for \( P_2 \) and \( Q_2 \)**:
   - We will evaluate each combination of \( X_{10} \) and \( Y_{10} \) to find which ones satisfy the conditions for \( P_2 \) and \( Q_2 \).

Let's compute \( P_2 \) and \( Q_2 \) for each combination.The valid combination that satisfies the conditions is:

- \( X_{10} = 163 \)
- \( Y_{10} = 89 \)

The corresponding values of \( P_2 \) and \( Q_2 \) are:

- \( P_2 = 1 \) (binary: `1`), which has a single 1 bit.
- \( Q_2 = 251 \) (binary: `11111011`), which has a single 0 bit.

### Summary
The correct values are:
- **\( X_{10} = 163 \)**
- **\( Y_{10} = 89 \)**

Would you like further details on the solution, or have any other questions? Here are some related questions to explore:

1. How do bitwise AND and OR operations work with binary numbers?
2. How do you identify prime numbers quickly?
3. What are the properties of binary numbers with a single 1 or 0 bit?
4. How can bitwise operations be applied in other mathematical problems?
5. What methods can be used to convert decimal numbers to binary and vice versa?

**Tip**: Understanding bitwise operations can be very useful in computer science, especially in tasks involving low-level data manipulation and optimization!

Explore how to find prime numbers X10 and Y10 satisfying specific conditions with binary operations. This problem involves identifying prime numbers of specific forms and evaluating bitwise AND and OR operations.

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