The problem asks for the integer solution to the Diophantine equation:

$4389x + 2919y = 231$

and to find the value of $x_0 - y_0$, where $(x_0, y_0)$ is the solution.

Solving the Diophantine Equation:

We are dealing with a linear Diophantine equation of the form:

$ax + by = c$

where $a = 4389$, $b = 2919$, and $c = 231$. The first step is to check if a solution exists. For a solution to exist, the greatest common divisor (gcd) of $a$ and $b$ must divide $c$. Let’s calculate the gcd of $4389$ and $2919$.

Step 1: Calculate $\gcd(4389, 2919)$ using the Euclidean algorithm.

1. $4389 \div 2919 = 1$ remainder $1470$ $\Rightarrow 4389 = 2919 \times 1 + 1470$
2. $2919 \div 1470 = 1$ remainder $147$ $\Rightarrow 2919 = 1470 \times 1 + 147$
3. $1470 \div 147 = 10$ remainder $0$ $\Rightarrow 1470 = 147 \times 10$

So, $\gcd(4389, 2919) = 147$.

Step 2: Check if $\gcd(4389, 2919)$ divides $231$.

Since $231 \div 147 = 1.57$, which is not an integer, this means that there is no integer solution to the equation $4389x + 2919y = 231$.

Conclusion:

Since $\gcd(4389, 2919)$ does not divide $231$, the equation has no integer solutions. Therefore, the value of $x_0 - y_0$ cannot be determined because the solution does not exist.

Would you like more information on the steps involved in the Euclidean algorithm?

5 Related Questions:

1. How do we solve linear Diophantine equations when the gcd divides the constant?
2. What is the significance of the gcd in determining whether a solution exists for Diophantine equations?
3. How can we extend the Euclidean algorithm to solve other types of equations?
4. Can you explain how to solve Diophantine equations using the extended Euclidean algorithm?
5. What are some real-world applications of Diophantine equations?

Tip: Always check if the gcd of the coefficients divides the constant term in a linear Diophantine equation to ensure the existence of integer solutions.