13. In what ways can you give 78 rubles, having only five-ruble and three-ruble banknotes?
13. Let $x$ be the number of 5-ruble stamps, and $y$ be the number of 3-ruble stamps, then $78 = 5x + 3y$, from which $x = 15 - \frac{3(y-1)}{5}$. Since $x$ is a positive integer, $15 - \frac{3(y-1)}{5} > 0$, and thus $y < 26$. Additionally, $\frac{3(y-1)}{5}$ is an integer, meaning $y-1$ is a multiple of 5. Let $y-1 = 5k$, $0 \leq k < 5$.

Answer. 78 rubles can be paid in five ways.

| $k$ | 0 | 1 | 2 | 3 | 4 |
| :--- | :--- | :--- | :--- | :--- | :--- |
| $y$ | 1 | 6 | 11 | 16 | 21 |
| $x$ | 15 | 12 | 9 | 6 | 3 |