The Pirates and the Monkey

There are five pirates, one monkey, and lots of coconuts on an island. The pirates are supposed to share the coconuts on the next day, but while everybody is sleeping, the first pirate gives 1 coconut to the monkey, splits the remaining coconuts into 5 equal piles, and secretly keeps one of the piles for himself. Later, the second pirate does the same, then the third one, the fourth one, and the fifth one. On the morning, the pirates wake up and split all the remaining coconuts in five, leaving one last coconut for the monkey.
What is a possible number for the number of coconuts on the island?

SOLUTION

Notice that if we find a certain number of coconuts which works, then we can add 5⁶ and get a new one. Now imagine the pirates start with -4 coconuts, i.e. they have a total loan of 4 coconuts. Every time a pirate wakes up, he gives 1 coconut to the monkey, which makes the total loan 5 coconuts. Then the pirate keeps a loan of 1 coconut for himself and leaves -4 coconuts. Now we just add 5⁶ coconuts to -4 to make the number positive and get 5⁶ – 4 coconuts as a possible answer.

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