Diophantine equation, how many solutions such that $x \ge 0$ and $y \ge 0$.

Question

I've been working on this problem for a while now. $\text{How many solutions (x, y) above are such that x ≥ 0 and y ≥ 0? for the diophantine equation: }$ $$ 473x + 338y = 4053. $$

I know how to solve diophantine equations so I'll jump to the part where I am unsure. We get the equation: $473 \cdot ( -5 \cdot 4053) + 338 \cdot ( 7 \cdot 4053) = 4053 $ The general solutions should be $x = -5 \cdot 4053 - 338n$ , $\;n \in \mathbb{Z}$ and

$y = 7 \cdot 4053 + 473n$ , $\,n \in \mathbb{Z}$

lets see where $x \ge 0$ and $y \ge 0$:

$-5 \cdot 4053 - 338n ≥ 0 \implies n ≤ -\frac{5 \cdot 4053}{338} = -59.955...$

$7 \cdot 4053 + 473n ≥ 0 \implies n ≥ -\frac{7 \cdot 4053}{473} = -59.980... $

Thus, $n$ cannot be an integer because there are no integers in the interval $[-59.955.. , -59.980..]$, therefore this Diophantine equation has no solutions for $x \ge 0$ and $y \ge 0$? Is this a correct conclusion?

---

Answer 1

@OldPeter posted a helpful comment and gave clarity to my question!