Problem 5. A group of tourists is dividing cookies. If they evenly distribute two identical packs, one extra cookie will remain. But if they evenly distribute three such packs, 13 extra cookies will remain. How many tourists are in the group? [7 points] (I.V. Raskina)
Answer: 23.

First solution. Distribute three times two packs, 3 * 1 = 3 cookies will remain. But the same six packs of cookies can be distributed differently - three and another three, and then 2 * 13 = 26 cookies will remain. Therefore, 26 - 3 = 23 cookies can be divided equally among the tourists. Since the number 23 is prime, this is only possible if there are 23 tourists.

Second solution. First, divide two packs of cookies. One extra cookie remains. Divide the third pack. It contains 13 - 1 = 12 extra cookies. But then there should have been 12 * 2 = 24 extra cookies in the two packs. Why is there actually only one? Because 24 - 1 = 23 cookies the tourists were able to divide equally. Since the number 23 is prime, this is only possible if there are 23 tourists.

Comment. The number of cookies in a pack cannot be determined from the problem statement. There could have been 12, 35, 58, etc.