Please help me solve this Diophantine Equation : $11x+13y$=$1000$ where $(x,y)$ belongs to positive integers .
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Do you want a single solution? All solutions? What have you tried? – Arthur Apr 04 '19 at 13:30
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This question appeared in a regional level olympiad and asked for all positive integral solutions of $ x , y $. – Rishabh Shetty Apr 04 '19 at 13:32
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Have you tried expressing $x$ in terms of $y$ (or vice versa) and plugging positive integers into $y$? – wjmolina Apr 04 '19 at 13:33
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I tried solving in a traditional way and found only one solution $(x,y)$= $(6000,-5000)$ and $-5000$ is not a positive integer – Rishabh Shetty Apr 04 '19 at 13:34
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No . Please provide me at least guidance to solve the equation – Rishabh Shetty Apr 04 '19 at 13:36
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google Linear Diophantine Equations – J. W. Tanner Apr 04 '19 at 13:39
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I know to solve a Diophantine equation but I am unable to find solution for this question – Rishabh Shetty Apr 04 '19 at 13:43
2 Answers
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Note that$$y=\frac{1}{13}(1000-11x).$$This is saying that $y$ is an integer when $1000-11x=13k$ for some integer $k$, i.e., when $12\equiv11x\pmod{13}$. This happens when $x\equiv7\pmod{13}$, i.e., when $x=7+13k$ for some integer $k$.
Since $x$ and $y$ have to be positive, $k\geq0$ and$$y=\frac{1}{13}(1000-11(7+13k))>0\implies k<\frac{71}{11}\implies k\leq6.$$So there are $7$ solutions and they are $(7,71)$, $(20,60)$, $(33,49)$, $(46,38)$, $(59,27)$, $(72,16)$, and $(85,5)$.
wjmolina
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Can you solve this in a linear diophantine equation and give me all solutions ? – Rishabh Shetty Apr 04 '19 at 13:49
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I will show you how to find a single solution. General solution can then be found by this. We know that (By Bezout's Lemma) there exist integers $k,l$ such that $$11k+13l=1$$ Now, by trial and error we can easily see that $k = 6$ and $l=-5$. Therefore, $$11(6000)+13(-5000)=1000$$ Now, use a little trick. We want to find positive solution and we are getting negative one for $13$, add and subtract the nearest multiple of $11$ to $5000$ (which is $5005 = 11\times455$), we get, $$11(6000)+13(-5005)+13(5)=1000$$ $$\Longrightarrow11(6000)-13(11)(455)+13(5)=1000$$ $$\Longrightarrow11(6000-13\times455)+13(5)=1000$$ $$\Longrightarrow11(85)+13(5)=1000$$ Hence, we get the required solution.
Hope it is helpful:)
