Solving the Linear Diophantine Equation 23x - 49y 179

Linear Diophantine equations are a fascinating and useful topic in number theory. They are equations of the form (ax by c), where (a), (b), and (c) are integers, and we seek integer solutions for (x) and (y). In this article, we delve into the method of solving the linear Diophantine equation (23x - 49y 179).

Introduction to Linear Diophantine Equations

A linear Diophantine equation is a linear equation where the solutions are required to be integers. These equations are named after the 3rd-century Alexandrian mathematician Diophantus, who studied such equations and their solutions.

Steps to Solve the Given Equation

The given equation is (23x - 49y 179). Let's solve this equation step by step using the Extended Euclidean Algorithm.

Step 1: Find a Particular Solution

The first step involves finding a particular solution to the equation using the Extended Euclidean Algorithm. We start by finding the greatest common divisor (gcd) of 23 and 49.

Step 1.1: Finding the gcd(23, 49)

49 2 * 23 3 23 7 * 3 2 3 1 * 2 1 2 2 * 1 0

The gcd(23, 49) is 1, since we eventually reach 1 as the remainder.

Next, we use the Extended Euclidean Algorithm to express 1 as a combination of 23 and 49:

1 3 - 1 * 2 2 23 - 7 * 3, substituting gives 1 3 - 1 * (23 - 7 * 3) 8 * 3 - 1 * 23 3 49 - 2 * 23, substituting gives 1 (49 - 2 * 23) - 1 * 23 8 * 49 - 17 * 23

Thus, we have (1 8 * 49 - 17 * 23). Now, we scale this solution by multiplying both sides by 179:

179 8 * 179 * 49 - 17 * 179 * 23

This provides us with a particular solution:

x0 -17 * 179 -3043

y0 8 * 179 1432

Step 2: General Solution

The general solution for a linear Diophantine equation (ax by c) is given by:

x x0 (frac{b}{text{gcd}(a, b)}t) y y0 - (frac{a}{text{gcd}(a, b)}t)

Since the gcd(23, 49) is 1, the general solution is simplified to:

x -3043 49t

y 1432 - 23t

Where (t) is any integer.

Alternative Method: Modular Arithmetic

Another method involves using modular arithmetic to find the solution. We can rewrite the equation:

49y -179 23x 49y -179 mod 23 -18 3y -18 mod 23 -6

Thus, y -6 mod 23 17 23k. Plugging this back into the original equation:

49(17 23k) -179 23x x 44 49k

Therefore, the general solution is:

x 44 49k

y 17 23k

Where (k) is an integer.

Conclusion

The complete solution to the equation 23x - 49y 179 is: