6.3. The hunter told a friend that he saw a wolf with a one-meter tail in the forest. That friend told another friend that a wolf with a two-meter tail had been seen in the forest. Passing on the news further, ordinary people doubled the length of the tail, while cowards tripled it. As a result, the 10th channel reported about a wolf with a tail 864 meters long. How many ordinary and how many cowardly people "grew" the wolf's tail? Provide all possible answers and prove that there are no others.
Solution: Note that when information is transmitted by ordinary people, the length of the tail is multiplied by 2, and when transmitted by cowardly people, it is multiplied by 3. Therefore, the number of twos in the product equals the number of ordinary people (and the number of threes equals the number of cowardly people) through whom the information has passed. This number does not depend on the order in which the information reached these people. Let's factorize the number 864 into prime factors. We get $864=2^{5} \cdot 3^{3}$. Therefore, the number of ordinary and cowardly people who grew the wolf's tail was 5 and 3, respectively.

Answer: 5 ordinary people and 3 cowardly people.

Recommendations for checking:

| is in the work | points |
| :--- | :--- |
| Correct and justified answer | 7 points |
| Answer is incorrect only due to arithmetic errors | 6 points |
| Correct answer is confirmed by an example showing how the wolf's tail length could have "changed" | 4 points |
| It is justified that the length of the "grown" tail does not depend on the order in which the information was transmitted by ordinary and cowardly people, but there is no solution. | 2 points |
| Correct answer without justification | 1 point |
| Principally incorrect reasoning | 0 points |