To find the number of integer solutions for the linear Diophantine equation \(17x + 43y = 4200\), we can follow these steps:

1. **Check for a solution existence condition**:
   A linear Diophantine equation \(ax + by = c\) has integer solutions if and only if the greatest common divisor (GCD) of \(a\) and \(b\) divides \(c\).

   For the equation \(17x + 43y = 4200\):
   \[
   \text{GCD}(17, 43) = 1
   \]
   Since 1 divides 4200, there are integer solutions.

2. **Find a particular solution**:
   Using the Extended Euclidean Algorithm, we can find a particular solution \((x_0, y_0)\) to the equation.

   However, for simplicity, we can test simple values by trial:
   By trying different values, one solution can be found: \((x_0, y_0) = (200, -100)\).

3. **General solution form**:
   The general solution for the equation is:
   \[
   x = x_0 + \frac{b}{\text{GCD}(a, b)}t \quad \text{and} \quad y = y_0 - \frac{a}{\text{GCD}(a, b)}t
   \]
   where \(t\) is an integer.

   For our case:
   \[
   x = 200 + 43t \quad \text{and} \quad y = -100 - 17t
   \]

4. **Determine the range of \(t\) for integer solutions**:
   We need to find the range of \(t\) for which both \(x\) and \(y\) remain integers.

   Since \(x = 200 + 43t\) must be positive, \(t\) must satisfy:
   \[
   200 + 43t > 0 \quad \Rightarrow \quad t > -\frac{200}{43} \approx -4.65
   \]
   Thus, \(t \geq -4\).

   Since \(y = -100 - 17t\) must be non-negative:
   \[
   -100 - 17t \geq 0 \quad \Rightarrow \quad t \leq -\frac{100}{17} \approx -5.88
   \]
   Thus, \(t \leq -6\).

   The integer values of \(t\) are \(t = -6\).

5. **Count the integer solutions**:
   Since \(t\) only takes one value (\(t = -6\)), there is exactly **1 integer solution**.

### Answer:
The equation \(17x + 43y = 4200\) has exactly **1 integer solution**.

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Would you like more details or have any questions? Here are some related questions:

1. How do you use the Extended Euclidean Algorithm to find a particular solution?
2. What happens if the GCD does not divide the constant term in a Diophantine equation?
3. Can you have more than one integer solution for this type of equation?
4. How would you solve a similar equation with different coefficients?
5. What if the equation had a negative constant term?
6. How do you determine the GCD of two numbers?
7. What is the significance of the general solution form in linear Diophantine equations?
8. How can you visualize solutions to Diophantine equations?

**Tip:** For Diophantine equations, always start by checking the GCD condition to see if integer solutions exist.

Explore how to find the number of integer solutions for the linear Diophantine equation 17x + 43y = 4200 using mathematical techniques. Understand the conditions for existence of integer solutions, find a particular solution, and derive the general solution form. Suitable for advanced high school students and enthusiasts of number theory and linear algebra.

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