## Task 3 - 170833

The fraction $\frac{9}{91}$ is to be represented as the difference of two positive proper fractions with denominators 7 and 13.

Investigate whether such a representation exists, whether there is more than one, and determine all such representations!
Assuming there is such a representation, it reads

$$
\frac{9}{91}=\frac{x}{7}-\frac{y}{13}
$$

where $x$ and $y$ are natural numbers with $0<x<7$ and $0<y<13$. Since $91=7 \cdot 13$, we have $13 x-7 y=9$ and thus $y=\frac{13 x-9}{7}$.

For $x=1, \ldots, 6$, we obtain:

| $x$ | $13 x$ | $13 x-9$ | $x$ | $13 x$ | $13 x-9$ |
| :---: | :---: | :---: | :---: | :---: | :---: |
| 1 | 13 | 4 | 2 | 26 | 17 |
| 3 | 39 | 30 | 4 | 42 | 43 |
| 5 | 65 | 56 | 6 | 78 | 69 |

Of the numbers listed in the third column, only the number 56 is divisible by 7. Therefore, we obtain an integer value for $y$ only for $x=5$, and specifically $y=8$.

Thus, the only representation that satisfies the conditions of the problem is

$$
\frac{9}{91}=\frac{5}{7}-\frac{8}{13}
$$

This representation meets the conditions of the problem; $\frac{5}{7}$ and $\frac{8}{13}$ are positive proper fractions, and (1) holds.