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I have a equation $74x+47y=2900$. I know that all the solutions are $x = 2900\cdot x_0+k\cdot47$ and $y = 2900\cdot y_0 - k\cdot74$, where $k\in\mathbb Z$ and $x_0, y_0$ are solution of $74x+47y=1$ (gcd(74,47) = 1). For example, $x_0=7$ and $y_0=-11$. But I am not sure is it enough to prove that there is no positive integer solution.

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We need $2900\cdot7+47k>0\iff k>\dfrac{2900\cdot7}{47}>-431.914894\implies k\ge-430$

and $2900\cdot(-11)-74k>0\iff k<-\dfrac{2900\cdot(-11)}{74}<-431.081081081\implies k\le431$

as $k$ is an integer

Or $-431.914894<k<-431.081081081$

But there is no integer $k$ in the range

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