It says use the Euclidean Algorithm method, but you must also use the extended Eucliadean method. 1) Use the Euclidean algorithm to find (752,677) and express it as a linear combination of theseintegers.2) Find all solutions to the linear diophantine equation 752x+677 y = 13.

Answered: 1) Use the Euclidean algorithm to find…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

It says use the Euclidean Algorithm method, but you must also use the extended Eucliadean method.

1) Use the Euclidean algorithm to find (752,677) and express it as a linear combination of these
integers.
2) Find all solutions to the linear diophantine equation 752x+677 y = 13.

Expert Solution

Step 1

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Introduction:

The following facts form the basis of the algorithm.

GCD remains the same if we lower a larger integer by subtracting a smaller one from it. Therefore, if we repeatedly deduct the larger of two, we arrive at GCD.
Now, if we divide the smaller number rather than subtract it, the algorithm halts when we reach the final result of 0.