11.4 The pirates, led by John Silver on Treasure Island, found Billy Bones' chest, which contained 40 coins worth 1 ducat each and 40 coins worth 5 ducats each. John Silver has not yet decided how to divide this money among all the pirates (he does not want to take anything for himself). For what maximum number of pirates can any of his decisions on dividing the coins be implemented (each pirate must receive a whole number of ducats; it is possible that some pirates will not receive any money at all)? Justify your answer.
Solution. We will show that if there are no more than 11 pirates, Silver can divide the coins as he wishes. Indeed, let the $i$-th pirate need to receive $S_{i}$ coins. $S_{i}=5 x_{i}+a_{i}$, for some integers $x_{i}$ and $a_{i}$ (where $a_{i}$ is the remainder of the division of $S_{i}$ by 5). Note that the sum of all $S_{i}$ is 240 ducats, so the sum of all $a_{i}$ is divisible by 5. Each of the numbers $a_{i}$ is no more than 4, so the sum of $a_{i}$ is no more than 44, and being divisible by 5, it is no more than 40. We will give out all $a_{i}$ in one-ducat coins - there will be enough. Now it remains to give each pirate a sum that is a multiple of 5 ducats. This is easily done since the number of one-ducat coins left will be a multiple of 5.

If there are 12 or more pirates, the captain will not be able to pay, for example, such sums: 10 pirates 4 ducats each, one 3 ducats, and one the rest, 197 ducats. Indeed, the first 11 pirates can only be paid with one-ducat coins, and for this, no less than 43 are needed.

Answer: 11 pirates.

| THERE IS IN THE SOLUTION | SCORE |
| :--- | :---: |
| Correct and justified answer | 7 points |
| Correct justification that for any number of pirates less than 12 and any decision by Silver, the distribution is possible plus an example of distribution for 12 people, which is not possible (if its impossibility is not proven) | 4 points |
| Correct justification that for any number of pirates less than 12 and any decision by Silver, the distribution is possible | 3 points |
| Correct example of when it will not be possible to distribute the coins for 12 pirates (with proof of impossibility) | 2 points |
| Correct example of when it will not be possible to distribute the coins for 12 pirates (without proof of impossibility) | 1 point |
| Correct answer without justification or with incorrect justification OR examples of the impossibility of distributing money for a number of pirates greater than 12 | 0 points |