767. Solve the equation in natural numbers $x$ and $y$

$$
65 x - 43 y = 2
$$
$\triangle$ First, let's solve this equation in integers using a familiar method. We will provide the corresponding formulas without derivation (you can verify them by substitution into the equation):

$$
x=4-43 t, \quad y=6-65 t
$$

Now, let's find all integer $t$ for which $x$ and $y$ are positive. To do this, we solve the system of inequalities in integers:

$$
\left\{\begin{array}{l}
4-43 t>0 \\
6-65 t>0
\end{array}\right.
$$

We get: $t<\frac{6}{65}$. The last inequality is satisfied by all non-positive integers $t$.

Answer: $x=4-43 t, \quad y=6-65 t$, where $t=0,-1,-2, \ldots$