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Can Anyone Help Me?

I am struck with the questions which include Diophantine equations. I have a lot of them(questions).The question asks :

Find the number of pairs (x,y) of positive integral solutions for the following equation.

\(2x+3y = 763\)

Please provide Solution.

Note by Vishal Yadav
2 years, 1 month ago

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What have you tried? Where did you get stuck?

Check out Linear Diophantine Equations.

Calvin Lin Staff - 2 years, 1 month ago

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Sir, isn't there a much time saving way ? Because finding the first random solutions takes a lot time even if you go by the Division Algorithm.

Vishal Yadav - 2 years, 1 month ago

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Not particularly.

For small numbers, you can guess. E.g. \( y = 1, x = \frac{760}{2} \) works. You can also just try \( x = 1, 2, 3, \ldots \) and hopes it works out (which it eventually will).

The Extended Euclidean Algorithm for an algorithm that guarantees finding the values within a reasonable amount of time. I think it is on the order of \( O ( \log n ) \), instead of testing all values which has the order of \( O(n) \).

Calvin Lin Staff - 2 years, 1 month ago

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Hint: \(2x + 3y = 2x + 3y - 6n + 6n = 2(x - 3n) + 3(y + 2n) = 763 \). And find the smallest and largest integer solution of \(x\) that satisfy the original equation.

Pi Han Goh - 2 years, 1 month ago

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