How to Find Integral Solutions to the Equation 17 cdot 19 x cdot 17 cdot 23 y cdot 19 cdot 23 z 3 cdot 3491

In this guide, we will walk through a detailed process to find all integral solutions to the equation:

17 cdot 19 x cdot 17 cdot 23 y cdot 19 cdot 23 z 3 cdot 3491

Factoring and Simplification

The first step in solving such problems is to factor all the numbers involved. Let's start by factorizing the given equation:

17 cdot 19 x cdot 17 cdot 23 y cdot 19 cdot 23 z 3 cdot 3491

Next, we take the common factors out of the first two terms and move the third term to the right-hand side (RHS):

17 x cdot 19 x cdot 23 y 3 cdot 3491 - 19 cdot 23 z

To make the RHS a multiple of 17, we rewrite the equation as:

17 x cdot 19 x cdot 23 y 17 cdot 616 - 25 z (1 - 12z)

This expression is a multiple of 17 if 12z - 1 is a multiple of 17. Since 12 times 10 equals 17 times 7 (1), z must be in the form z 10 17m, where m is an integer.

Substituting z 10 17m into the equation:

19 x cdot 23 y 359 - 437m

Finding Specific Solutions

To find one solution, we need to find a solution to 19 x cdot 23 y 1. This can be done using the extended Euclidean algorithm, which provides a method to find the greatest common divisor (GCD) of two numbers and express it as a linear combination of those numbers.

A solution for 19 x cdot 23 y 1 is x -6 and y 5. Multiplying these by the RHS of the original equation:

x 2622m - 2154

y -2185m 1795

Generalized Solution

We can now generalize the solution by taking multiples of 23 from x and the same multiples of 19 from y without changing the result. By taking multiples of 114m - 94, we get:

x 8

y 9 - 19m

Adding multiples of n gives us the general solution:

x 8 23n

y 9 - 19mn

z 10 17m

Conclusion

By following these steps, we have successfully derived the general solution to the given equation. This method can be applied to similar problems to find integral solutions. Understanding and applying the extended Euclidean algorithm is crucial for solving such equations efficiently.

Keywords: Solving Equations, Integral Solutions, Extended Euclidean Algorithm