Solving Linear Diophantine Equations: Methods and Examples

Solving linear Diophantine equations involves finding integer solutions to equations of the form Ax By C. This article will cover the step-by-step processes to solve such equations, with a particular focus on methods such as the Extended Euclidean Algorithm and modular arithmetic techniques. We will also illustrate these concepts with several examples, including the equation 73x 151y 3.

Introduction to Linear Diophantine Equations

A linear Diophantine equation is a type of equation where the coefficients are integers, and the solutions are required to be integers. These equations are commonly represented as Ax By C, where A, B, and C are integers, and x and y are the variables to be solved for. The equation 73x 151y 3 is a specific instance of such an equation.

Solving the Equation (73x 151y 3)

The first step is to verify if an integer solution exists. This can be done by checking if the greatest common divisor (GCD) of the coefficients A and B divides the constant term C.

Step 1: Verify the GCD

First, note that gcd(73, 151) 1, which means that 73 and 151 are coprime. Since 1 divides 3, an integer solution exists.

Step 2: Solve Using Modular Arithmetic

Next, we solve the equation modulo 73:

151y 3 mod 73 151 equiv 5 mod 73, thus simplifying to 5y 3 mod 73 75y 45 mod 73, further simplifies to 2y 118 mod 73, which gives y 59 mod 73.

Therefore, y 59 73k for any integer k.

Step 3: Substitute and Solve for (x)

Now, we substitute y 59 73k into the original equation:

73x - 151(59 73k) 3

This simplifies to x -122 - 151k. The general solution is:

x -122 - 151k

y 59 73k

Another Example: Solving (73x - 151y 3)

Step 1: Analyze the GCD

The GCD of 73 and 151 is 1 again, confirming that an integer solution exists.

Step 2: Solve for a Specific Solution

We start by solving the equation 73x - 151y 1:

(73 - 2 cdot 151 -229 151 1)

Multiplying through by 3 gives the specific solution:

73 cdot 180 - 151 cdot 87 3.

The particular solution is:

x 180

y -87

Final Example: A More Complex Equation

Consider the equation 72x - 150y - x - y 3. Simplifying this, we get:

24x - 50y - x - y 1

This can be further simplified to:

151y - 26x 1

By solving, we find:

x 6p - 5p - 1/26

Substituting and solving further, we get:

x 151n - 29

y -73n - 14

For any integer n.

Conclusion

By utilizing the Extended Euclidean Algorithm and modular arithmetic, one can systematically find integer solutions to linear Diophantine equations. The process involves verifying the GCD, solving reduction cases, and substituting back into the original equation. This set of examples highlights the practical application of these techniques in solving diverse Diophantine equations.