The answer is 195.

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Rearranging $23x + 43y = 3000$ gives $y = \frac{3000 - 23x}{43}$ which can be further rearranged to $y = 70 - \frac{10}{43} - x + \frac{20x}{43}$ and again to $y = 70 - x + 10(\frac{2x - 1}{43})$ . For $y$ to be a natural number, $2x - 1$ must be divisible by $43$ .

Since $x$ is a natural number, $2x - 1$ must be odd, and so $\frac{2x - 1}{43}$ must be odd, and so $2x - 1 = 43(2n + 1)$ for any natural number $n$ , which simplifies to $x = 43n + 22$ .

When $y = 0$ , $23x + 43 \cdot 0 = 3000$ solves to $\frac{3000}{23} \approx 130.4$ . Since $x$ is a natural number it must be an integer between $0$ and $130$ , and the only values of $x = 43n + 22$ for any natural number $n$ in this range are $22$ , $65$ , and $108$ , which add up to $22 + 65 + 108 = \boxed{195}$ .