On July 1, 2010, Mrs. Hovorkova had a credit of 157.0 Kč on her mobile phone. The credit is gradually debited for calls, with 4.5 Kč deducted for each started minute. Mrs. Hovorkova does not send text messages and does not use any other paid services. She tops up her credit as needed, always by an amount of 400 Kč. On December 31, 2010, her credit was 353.0 Kč. How many times at least did Mrs. Hovorkova top up her credit during the mentioned half-year?

(L. Simünek)
The amount in crowns that Mrs. Hovorkova called during the period from July 1 to December 31 is a whole number because the initial and final credit balances and the top-up amounts are always whole numbers. For each minute started, 4.5 Kč is charged, and to reach a whole number, an even number of minutes must have been charged. Therefore, the amount called during this period is divisible by nine without a remainder.

Since the initial credit is less than the final credit, Mrs. Hovorkova must have topped up at least once during the observed period. If she topped up only once, the amount called would be \(157 + 400 - 353 = 204\) (Kč). By the digit sum of the number 204, we see that it is not a multiple of nine, so we reject this option. If she topped up twice, the amount called would be 400 Kč more, i.e., 604 Kč. We reject this option as well because 604 is not a multiple of nine. Similarly, we reject the option of three top-ups leading to an amount of 1004 Kč. If she topped up four times, the amount called would be 1404 Kč. This is possible because 1404 is a multiple of nine. Therefore, Mrs. Hovorkova topped up at least four times during the observed period.

Another solution. It is clear that Mrs. Hovorkova topped up her credit at least once. In the table, we will consider different numbers of top-ups: each time we will determine the amount called during the observed period and the number of minutes this amount corresponds to. We will continue until the number of charged minutes is a whole number.

| number of top-ups | amount called in Kč | number of charged minutes |
| :---: | :---: | :---: |
| 1 | \(157 + 400 - 353 = 204\) | \(204 \div 4.5 \approx 45.3\) |
| 2 | \(157 + 2 \cdot 400 - 353 = 604\) | \(604 \div 4.5 \approx 134.2\) |
| 3 | \(157 + 3 \cdot 400 - 353 = 1004\) | \(1004 \div 4.5 \approx 223.1\) |
| 4 | \(157 + 4 \cdot 400 - 353 = 1404\) | \(1404 \div 4.5 = 312\) |

Mrs. Hovorkova topped up at least four times during the observed period.

Evaluation. 4 points for verifying the options 204, 604, 1004, 1404; 1 point for explaining why the options are rejected; 1 point for the correct conclusion.