Solving for Integer Pairs Using the Extended Euclidean Algorithm: A Comprehensive Guide

The Extended Euclidean Algorithm is one of the most powerful tools for solving linear Diophantine equations. It provides a method to find not just one, but an infinite set of solutions to the equation (ax by 1), where (a) and (b) are integers and the greatest common divisor (gcd) of (a) and (b) is 1.

Introduction to the Extended Euclidean Algorithm

The Euclidean algorithm is a well-known method for finding the greatest common divisor (gcd) of two integers, (a) and (b). The extended version of this algorithm not only finds the gcd but also provides coefficients (x) and (y) such that (ax by gcd(a, b)).

Solving an Example with the Extended Euclidean Algorithm

Consider the equation (373x 337y 1). Our goal is to find a pair of integers ((x, y)) that satisfies this equation. Let's solve this step-by-step using the Extended Euclidean Algorithm.

Step-by-Step Solution

1. First, we apply the Euclidean method to find the gcd (and the coefficients) of 373 and 337.

373 ÷ 337 1 - remainder 36

337 ÷ 36 9 - remainder 13

36 ÷ 13 2 - remainder 10

13 ÷ 10 1 - remainder 3

10 ÷ 3 3 - remainder 1

3 ÷ 1 3 - remainder 0

The gcd is 1, and the steps of the algorithm provide the coefficients for (373x 337y 1).

Reconstructing from the remainder operations:

(1 3 - 2 times 3)

(1 3 - 2 times (10 - 3 times 3) 7 times 3 - 2 times 10)

(1 7 times (13 - 10) - 2 times 10 7 times 13 - 9 times 10)

(1 7 times 13 - 9 times (36 - 13 times 2) 25 times 13 - 9 times 36)

(1 25 times (337 - 36 times 9) - 9 times 36 25 times 337 - 236 times 36)

(1 25 times 337 - 236 times (373 - 337) 261 times 337 - 236 times 373)

Thus, one solution is (x -236), (y 261).

Verification and Multiple Solutions

To verify, substitute the values into the original equation: (-236 times 373 261 times 337 1). This confirms the solution.

Since (373) and (337) are coprime, there are infinitely many solutions. You can generate other solutions using the general form: (x -236 337k), (y 261 - 373k), where (k) is any integer.

Another Example

Now, let's solve the equation (1 37 337y).

Step-by-Step Solution

1. First, we find the gcd using the Euclidean method:

370 ÷ 337 1 - remainder 33

337 ÷ 33 10 - remainder 7

33 ÷ 7 4 - remainder 5

7 ÷ 5 1 - remainder 2

5 ÷ 2 2 - remainder 1

2 ÷ 1 2 - remainder 0

The gcd is 1, and we have the equation (1 2 - 2 times 1). Working through the remainders:

(1 2 - 2 times (5 - 2 times 2) 5 times 2 - 2 times 5)

(1 5 times (7 - 5) - 2 times 5 5 times 7 - 7 times 5)

(1 5 times 7 - 7 times (33 - 10 times 7) 75 times 7 - 10 times 33)

(1 75 times (370 - 337) - 10 times 337 75 times 370 - 85 times 337)

Thus, one solution is (x 75), (y -85).

The general form for all solutions is (x 75 337k), (y -85 - 370k).

Conclusion

The Extended Euclidean Algorithm is a powerful tool for solving linear Diophantine equations and finding integer pairs. By understanding and applying this method, you can generate multiple solutions to such equations.