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The Coconuts Problem

Five sailors survive a shipwreck and swim to a tiny island where there is nothing but a coconut tree and a monkey. The sailors gather all the coconuts and put them in a big pile under the tree. Exhausted, they agree to go to wait until the next morning to divide up the coconuts.
At one o'clock in the morning, the first sailor wakes. He realizes that he can't trust the others, and decides to take his share now. He divides the coconuts into five equal piles, but there is one left over. He gives that coconut to the monkey, buries his coconuts, and puts the rest of the coconuts back under the tree.

At two o'clock, the second sailor wakes up. Not realizing that the first sailor has already taken his share, he too divides the coconuts up into five piles, leaving one over which he gives to the monkey. He then hides his share, and piles the remainder back under the tree.

At three, four and five o'clock in the morning, the third, fourth and fifth sailors each wake up and carry out the same actions.

In the morning, all the sailors wake up, and try to look innocent. No one makes a remark about the diminished pile of coconuts, and no one decides to be honest and admit that they've already taken their share. Instead, they divide the pile up into five piles, for the sixth time, and find that there is yet again one coconut left over, which they give to the monkey.

How many coconuts were there originally?

*Taken from
http://www.psc.edu/~burkardt/puzzles/coconut_puzzle.html.*