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Question:

How many integer solutions (consisting of two integers x and y) does the equation below have? Note: You are not required to find any solutions if there are any.

$ 294x + 266y = 28$

My attempt:

I am not sure what the question means by "how many". We know that gcd$(294,266)$ divides $28$ so there are infinitely many solutions to this Diophantine equation.

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    $\begingroup$ Well, that's your answer then. $\endgroup$ – Ivan Neretin Aug 9 '17 at 7:24
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Yes, ther are infinite solutions.

Given $294x + 266y = 28$, we can see that $\gcd(294, 266) = 14$ divides the answer, $28$, so solutions are feasible.

Dividing the equation through by $14$, we get $21x+19y=2$. Clearly $(x,y) = (1,-1)$ is a solution and for any solution $(x,y)=(k,\ell)$ we will have $(k+19, \ell-21)$ and $(k-19, \ell+21)$ will also be solutions. Hence there are an infinite number of solutions.

In general after the division we will have coprime coefficients so using Bézout's identity we can adjust the values of $x$ and $y$ to give any integer answer and then generate the infinite set similarly to above.

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