4. Flint has five sailors and 60 gold coins. He wants to distribute them into wallets, and then give the wallets to the sailors so that each gets an equal number of coins. But he doesn't know how many sailors will be alive by the time of the distribution. Therefore, he wants to distribute the coins in such a way that they can be evenly divided among two, three, four, or five sailors. What is the minimum number of wallets he will need? Don't forget to prove that the number you found is the smallest.
Solution. Answer: 9 wallets. Example: $12,12,8,7,6,5,4,3,3$.

We will prove that 8 wallets are insufficient.

1) Note that each wallet should contain no more than 12 coins. Therefore, 15 coins must be made up of at least two wallets. This means that when we divide 8 wallets among four pirates, each should receive two wallets. Thus, they form four pairs with a sum of 15.
2) When dividing among five pirates, at least two pirates will receive one wallet each. Therefore, there are two wallets with 12 coins.
3) From point 1, it follows that these two wallets must be paired with two wallets containing 3 coins.
4) If there are only two wallets with 12 coins, the rest must form pairs with a sum of 12. Then, in addition to each wallet with 3 coins, there must be a wallet with 9 coins. And then each of them needs a pair with 6 coins (to make a sum of 15). Thus, the set of wallets is: $12,12,3,3,9,9,6,6$. Clearly, 20 coins cannot be obtained with these (20 is not divisible by 3).
5) If there are more than two wallets with 12 coins, i.e., at least three, then there must be at least three wallets with 3 coins (which complement them to 15 coins, see point 1). However, from such a set of wallets, it is impossible to form even one portion of 20 coins; to be able to get three such portions, at least three more wallets need to be added, then there will be no fewer than nine.