Linear Diophantine equations are equations of the form ax + by = c, where a, b and c are integers. These equations can be solved using a method known as the extended Euclidean algorithm. In this article, we will explain how to solve a linear Diophantine equation using the extended Euclidean algorithm.

Let’s consider the following example: 5x + 7y = 12. We want to find integer solutions (x,y) to this equation.

Step 1: Find the greatest common divisor of a and b

In our example, a = 5 and b = 7. To find the greatest common divisor of a and b, we use the Euclidean algorithm. The algorithm involves repeated division of the larger number by the smaller number, until the remainder becomes zero.

7 = 1*5 + 2

5 = 2*2 + 1

2 = 2*1 + 0

The remainder becomes zero when we divide 1 by 2. Therefore, the greatest common divisor of 5 and 7 is 1.

Step 2: Find the particular solution

To find a particular solution to the equation, we use the extended Euclidean algorithm. The algorithm involves solving the equation ax + by = gcd(a,b) for x and y.

In our example, we have to solve 5x + 7y = 1. We can use the Euclidean algorithm in reverse to find the coefficients of x and y that satisfy this equation.

1 = 5 – 2*2

1 = 5 – (7 – 5*1)*2

1 = 3*5 – 2*7

Therefore, x = -2 and y = 3 is a particular solution to the equation 5x + 7y = 1.

Step 3: Find the general solution

To find the general solution, we add an integer multiple of b/gcd(a,b) to x and an integer multiple of a/gcd(a,b) to y. In other words, we have:

x = -2 + 7k

y = 3 – 5k

where k is an integer.

To check if these solutions are valid for the original equation 5x + 7y = 12, we substitute x and y into the equation and solve for k.

5(-2 + 7k) + 7(3 – 5k) = 12

Simplifying this equation, we get:

28k – 16 = 0

k = 16/28 = 4/7

Since k is not an integer, there are no integer solutions to the equation 5x + 7y = 12.

Therefore, the linear Diophantine equation 5x + 7y = 12 has no integer solutions.

In conclusion, we have explained how to solve a linear Diophantine equation using the extended Euclidean algorithm. The method involves finding the greatest common divisor of a and b, finding a particular solution to the equation ax + by = gcd(a,b), and finding the general solution by adding an integer multiple of b/gcd(a,b) to x and an integer multiple of a/gcd(a,b) to y. However, not all linear Diophantine equations have integer solutions. Therefore, it is important to check if the solutions obtained are valid for the original equation.

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