The question in the image involves a linear Diophantine equation of the form $ax + by = c$, where $a$ and $b$ are nonzero integers, and $c$ is some integer. You are being asked to determine which of the given statements must be true about such an equation.

I'll go through each statement to provide an explanation of whether it's true or false.

1. "If the Diophantine equation has a rational solution, i.e., a choice of rational numbers $x = x_0, y = y_0$ that satisfies the equation, then the equation has at least one integer solution."

   • This statement is false. The existence of a rational solution does not imply that there must be an integer solution. For a Diophantine equation, we are only concerned with integer solutions, and rational solutions can exist even when no integer solutions do.

2. "By Bezout's identity, the Diophantine equation only has a solution if $c$ is equal to the greatest common divisor of $a$ and $b$."

   • This statement is false. According to Bezout's identity, the equation has a solution if and only if the greatest common divisor (gcd) of $a$ and $b$ divides $c$. $c$ doesn't have to be equal to the gcd, but it must be divisible by the gcd.

3. "All of the solutions of the Diophantine equation are of the form $x = x_0 + bk, y = y_0 - ak$, where $x = x_0, y = y_0$ is a solution."

   • This statement is true. If $x_0, y_0$ is a particular solution to the equation, then all integer solutions can be expressed in the form $x = x_0 + bk, y = y_0 - ak$, where $k$ is any integer. This is a standard result for linear Diophantine equations.

4. "If $a, b$ have a common divisor that does not divide $c$, then there are no solutions."

   • This statement is true. For the equation $ax + by = c$ to have a solution, the greatest common divisor of $a$ and $b$ must divide $c$. If it doesn't, there are no integer solutions.

5. "If the greatest common divisor of $a$ and $b$ is also a divisor of $c$, then there exists at least one solution."

   • This statement is true. By Bezout's identity, if $gcd(a, b)$ divides $c$, then the Diophantine equation has at least one integer solution.

6. "If $a, b$ have a common divisor that divides $c$, then there must be at least one solution."

   • This statement is essentially a rephrasing of the previous one and is true for the same reason. If the gcd of $a$ and $b$ divides $c$, the equation has a solution.

7. "If the greatest common divisor of $a$ and $b$ is also a divisor of $c$, then there must exist infinitely many solutions."

   • This statement is true. Once you have one particular solution, you can generate infinitely many solutions using the formula $x = x_0 + bk, y = y_0 - ak$, where $k$ is any integer.

Summary of True Statements:

• Statement 3: True
• Statement 4: True
• Statement 5: True
• Statement 6: True
• Statement 7: True

Would you like further details on any specific statement or a step-by-step explanation? Here are five related questions you might consider next:

1. How does Bezout's identity help in solving Diophantine equations?
2. Can there be negative integer solutions to a linear Diophantine equation?
3. What is the geometric interpretation of the solutions to a Diophantine equation?
4. How can you find the particular solution to $ax + by = c$?
5. Why is the general solution expressed in the form $x = x_0 + bk$ and $y = y_0 - ak$?

Tip: For linear Diophantine equations, finding the gcd of the coefficients is key to determining whether solutions exist.