Jirka and Petr started their stopwatches and within 15 seconds of starting, each of them began to clap. Jirka clapped regularly every 7 seconds, Petr clapped regularly every 13 seconds. At the 90th second, both clapped simultaneously. Determine all possibilities for when Jirka and Petr could have started clapping.

(L. Dedková)
We will gradually list from the end, in which second Jirka and Petr clapped.

- Jirka: 90, 83, 76, 69, 62, 55, 48, 41, 34, 27, 20, **13**, **6**.
- Petr: 90, 77, 64, 51, 38, 25, 12.

The time of the first clap should be less than or equal to 15 seconds. Therefore, Jirka could have started clapping in the sixth or thirteenth second, while Petr had to start clapping in the twelfth second.

Another solution. The same result can be achieved using division with a remainder.

- Since 90 : 7 is 12, remainder 6, it means that Jirka could have started clapping in the 6th second (and clapped twelve more times by the 90th second).
- Since 90 : 13 is 6, remainder 12, it means that Petr could have started clapping in the 12th second (and clapped six more times by the 90th second).

We will gradually list the next claps of both boys and highlight those that meet the condition in the problem:

- Jirka: 6, 13, 20, ..
- Petr: 12, 25, . .

Evaluation. 3 points for the list and answer for each boy. If the solver forgets Jirka's second solution, deduct one point.