Task B-1.8. Determine all three-digit natural numbers that are divisible by 7, and when divided by 9 give a remainder of 5.
First solution. From the conditions in the problem, we have

$$
n=7 x \quad \text{and} \quad n=9 y+5, \quad x, y \in \mathbb{N}
$$

We solve the Diophantine equation $7 x=9 y+5$, or

$$
7 x-9 y=5
$$

One of its solutions is $x_{0}=2$ and $y_{0}=1$.

Then the general solution is $x=2+9 t, y=1+7 t, t \in \mathbb{N}$.

Since $100<n<1000$, and $n=7 x=14+63 t$, we have

$$
\begin{array}{r}
100<14+63 t<1000 \\
86<63 t<986 \\
1.3<t<15.6 \\
t \in\{2,3,4, \ldots, 15\}
\end{array}
$$

Thus, $n=14+63 t, t \in\{2,3,4, \ldots, 15\}$, or written out,

$$
n \in\{140,203,266,329,392,455,518,581,644,707,770,833,896,959\}
$$