Using your Head is Permitted

In the middle of the night, one of the party wakes up hungry. He decides not to wait for morning and to eat his share immediately. He divides the bananas into n equal batches, but in order for the equality to be perfect he gives the monkey one banana. He then eats his share and puts all n-1 remaining batches back in one pile.

After he falls back to sleep, another of the party wakes up. He, too, decides to eat his share immediately, and goes through exactly the same process as the first, including the giving of one banana to the monkey for the numbers to divide properly.

After the second man falls asleep, a third wakes up and repeats the same process. Eventually, each person in the party wakes up once during the night, gives the monkey a single banana and eats 1/n of the remaining bananas. Altogether, the monkey eats n bananas during the night.

Come morning, none of the people in the party remember the events of the previous night. They divide the bananas evenly, each getting 1/n of the remaining bananas and give one banana to the monkey for the numbers to divide properly.

The question: what, as a function of n, are all the possibilities for the number of bananas in the original bunch? Prove your answer.

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