Solving Linear Diophantine Equations and Finding Their Integral Solutions

Linear Diophantine equations are equations of the form ax by c, where the solutions x and y must be integers. These equations are crucial in number theory and various applications, including cryptography and computer science. In this article, we will explore how to find all integral solutions to the linear Diophantine equation 1124x - 84y gcd(1124, 84).

Introduction

Before solving the given Diophantine equation, it is important to find the greatest common divisor (GCD) of the coefficients 1124 and 84. Since gcd(1124, 84) 4, we can modify the equation to 281x - 21y 1 by dividing the entire equation by 4.

Step-by-Step Solution

Let's start by finding the GCD of 1124 and 84 using the Euclidean algorithm:

1. 1124 4 * 281

2. 84 4 * 21

3. gcd(1124, 84) 4

Now, we have the simplified equation: 281x - 21y 1. To solve this equation, we will use the method of finding the coefficients of x and y using the extended Euclidean algorithm.

The Euclidean algorithm gives us:

1. 281 - 13 * 21 8

2. 21 - 2 * 8 5

3. 8 - 1 * 5 3

4. 5 - 1 * 3 2

5. 3 - 1 * 2 1

By back-substitution, we find the coefficients:

1 3 - 1 * (5 - 1 * 3) 2 * 3 - 1 * 5 2 * (8 - 1 * 5) - 1 * 5 2 * 8 - 3 * 5 2 * 8 - 3 * (21 - 2 * 8) 8 * 8 - 3 * 21 8 * (281 - 13 * 21) - 3 * 21 8 * 281 - 109 * 21

Hence, one solution to the equation 281x - 21y 1 is x 8 and y -107.

The general solution for the equation can be written as:

x 8 21k

y -107 - 281k

where k is an integer.

General Solution and Verification

The generalized form of the solution for the equation 1124x - 84y 4 is:

x 8 21k

y -107 - 281k

Verify that this solution satisfies the original equation by substituting the general form back into the equation:

1124(8 21k) - 84(-107 - 281k) 1124 * 8 - 84 * 107 1124 * 21k 84 * 281k 8992 - 8992 23604k 23604k 4

This confirms that the generalized solution is correct.

Code Implementation

Here is a JavaScript code to find all integral solutions to the linear Diophantine equation 1124x - 84y 4:

for(let x  -200; x  200; let x   1) { 
    // Calculate Num
    let Num  1124 * x - 4;
    // Check if Num is divisible by 84
    if (Num % 84  0) {
        // Calculate y and print it
        let y  Num / 84;
        console.log(`x  ${x}, y  ${y}`);
    }
}

This code iterates through possible values of x and checks if the equation is satisfied. It prints the corresponding values of x and y.

Thus, we have successfully found all integral solutions to the given linear Diophantine equation.