# Prove that there is no positive integer solutions

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.

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 $$-431.914894
But there is no integer $$k$$ in the range