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Determine all solutions of the following Diophantine equations: 1335x801y = 534. Further, find all pairs, if any, which satisfy 1000 < x < 1050.

Determine all solutions of the following Diophantine equations: 1335x801y = 534. Further, find all pairs, if any, which satisfy 1000 < x < 1050.

Advanced Engineering Mathematics
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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Determine all solutions of the following Diophantine equations: 1335x - 801y = 534. Further, find all pairs, if any, which satisfy 1000 < x < 1050.
Transcribed Image Text:Determine all solutions of the following Diophantine equations: 1335x - 801y = 534. Further, find all pairs, if any, which satisfy 1000 < x < 1050.
Expert Solution
Step 1: Finding the complete solution

Applying Euclidean algorithm GCD1335,-801

-801 = (-1)×1335 + 534   1335 = 2×534 + 267    534 = 2×267 + 0

Therefore GCD1335,-801=2

Since 2|534 , the equation has integer solutions. Working the Euclidean algorithm backwards, we have that:

267 = (1 ×1335) + (-2 × 534)         = (-2 ×-801) + (-1 ×1335)

Now multiply both side by 2  we obtain 534

534=-4×-801+-2×1335

So one solution is x=-2, y=-4

All the solution is given by

x=-2+n801267⇒x=-2+3ky=-4+n1335267⇒y=-4+5n

Hence the complete solution is x=-2+3n, y=-4+5n

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