Mom paid 2700 Kč in a bookstore. She paid with two types of banknotes, 200 Kč and 500 Kč, and exactly. How many of each banknote could she have used? List all possibilities.

(M. Krejčová)

Hint. How many 500 Kč banknotes can be used so that the remaining amount can be paid with 200 Kč banknotes?
It is evident that the mother must have used fewer than 6 five-hundred-crown notes, because $6 \cdot 500 = 3000$ (Kč). We will sequentially examine all possibilities, i.e., that she used five, four, etc., down to no five-hundred-crown notes. Then we will determine how much she would have paid and how much more she would need to pay. Finally, we will decide whether the remaining amount could be paid with only two-hundred-crown notes. We will express the entire discussion in a table:

| number of 500Kč notes | 5 | 4 | 3 | 2 | 1 | 0 |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: |
| paid | 2500 | 2000 | 1500 | 1000 | 500 | 0 |
| remaining to pay | 200 | 700 | 1200 | 1700 | 2200 | 2700 |
| number of 200Kč notes | 1 | - | 6 | - | 11 | - |

The mother could pay in three ways:

- 5 five-hundred-crown notes and 1 two-hundred-crown note,
- 3 five-hundred-crown notes and 6 two-hundred-crown notes,
- 1 five-hundred-crown note and 11 two-hundred-crown notes.

Note. Similarly, we can proceed with the two-hundred-crown notes, of which there must be fewer than $14 (14 \cdot 200 = 2800)$; the corresponding table in this case would be significantly longer.