How can $768 \mathrm{Ft}$ in stamp duty be paid using $1.5 \mathrm{Ft}$ and $10 \mathrm{Ft}$ stamps, if the $5 \mathrm{Ft}$ stamps used are twice as many as the other two combined?
Let $x, y$ denote the number of 1 and 10 Ft stamps, respectively. According to the problem, $2(x+y)$ is the number of 5 Ft stamps, thus

$$
x + 5 \cdot 2(x+y) + 10 y = 768
$$

which simplifies to the indeterminate equation

$$
11 x + 20 y = 768
$$

We need to find the positive integer solutions of this equation.

$$
\begin{gathered}
x = \frac{768 - 20 y}{11} = 70 - 2 y + \frac{2 y - 2}{11} = 70 - 2 y + 2 \frac{y - 1}{11} = 70 - 2 y + 2 u \\
y = 11 u + 1 \\
x = 70 - 22 u - 2 + 2 u = 68 - 20 u \\
2(x + y) = 2(69 - 9 u) = 138 - 18 u
\end{gathered}
$$

From (1) $\quad 11 u + 1 > 0, \quad u > -\frac{1}{11} > -1$

From (2) $\quad 68 - 20 u > 0, \quad u < \frac{68}{20} = \frac{17}{5} < 4$

From (3) $\quad 138 - 18 u > 0, \quad u < \frac{138}{18} = \frac{23}{3} < 8$

(3) does not provide new information, so

$$
-1 < u < 4
$$

which means

$$
u = 0, 1, 2, 3
$$

and accordingly, we get 4 solutions:

| $x$ | 68 | 48 | 28 | 8 |
| ---: | :---: | :---: | :---: | :---: |
| $y$ | 1 | 12 | 23 | 34 |
| $2(x + y)$ | 138 | 120 | 102 | 84 |

Tibor Harza (Székesfehérvár, József Attila g. II. o. t.)