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We are given the equation \( 2002x + 923y = k \) and the condition that the greatest common divisor (gcd) of 2002 and 923 is 13, i.e.,

\[
\gcd(2002, 923) = 13.
\]

This tells us that any solution \(x\) and \(y\) in integers will exist if and only if the right-hand side of the equation \(k\) is divisible by the gcd, 13. In other words, \(k\) must be divisible by 13 for the equation to have integer solutions.

Now, let's check each option:

- **A) \( 2002x + 923y = 52 \):**
  - \( 52 \div 13 = 4 \), which is an integer.
  - Therefore, this equation **has solutions**.

- **B) \( 2002x + 923y = 26 \):**
  - \( 26 \div 13 = 2 \), which is an integer.
  - Therefore, this equation **has solutions**.

- **C) \( 2002x + 923y = 27 \):**
  - \( 27 \div 13 \approx 2.08 \), which is not an integer.
  - Therefore, this equation **does not have solutions**.

- **D) \( 2002x + 923y = 51 \):**
  - \( 51 \div 13 \approx 3.92 \), which is not an integer.
  - Therefore, this equation **does not have solutions**.

### Summary:
The equations that have integer solutions are:
- **A) 2002x + 923y = 52**
- **B) 2002x + 923y = 26**

Would you like further details on the process, or do you have any additional questions?

### Related questions:
1. How is the greatest common divisor (gcd) used in solving Diophantine equations?
2. What methods can be used to compute the gcd of two numbers like 2002 and 923?
3. Can all linear Diophantine equations be solved using the gcd approach?
4. How would the solution change if the gcd of 2002 and 923 was not 13?
5. What are some applications of linear Diophantine equations in real-world problems?

**Tip:** To check whether a Diophantine equation has solutions, always test if the constant term is divisible by the gcd of the coefficients of the variables.

Given that gcd(2002, 923) = 13, select all of the following equations that have solutions x, y ∈ Z: A) 2002x + 923y = 52, B) 2002x + 923y = 26, C) 2002x + 923y = 27, D) 2002x + 923y = 51.

This problem explores solving Diophantine equations, where gcd(2002, 923) = 13. Learn how to determine which equations have integer solutions by applying the divisibility rule and Bézout's identity.

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