Martin Gardner has died

Martin Gardner was a great puzzler. He wrote a column for Scientific American for many years about recreational mathematics. This Mechanical Puzzle Correspondent spent many hours on the following problem:

Five sailors are shipwrecked on a desert island. They quickly determine
that the only other inhabitant of the island is a monkey and that the only
food is coconuts. They set about collecting as many coconuts as they can
and put them all in a pile. By nightfall they are too tired to divide the
harvest; so they agree to go to sleep and divvy up the coconuts the next
morning.
During the night one sailor awakens, suspicious that the others might try
to cheat him, and desides to take his portion then and there and not wait
until morning. He divides the coconuts into five piles and finds there is
one coconut left over, which he gives to the monkey. He hides one of the
five piles, then puts the rest of the nuts together and returns to sleep.
About an hour later a second sailor awakens with the same suspicions and
does the same thing: He divides the coconuts into five piles, leaving one
extra, which he gives to the monkey. Then he hides what he thinks is his
share and goes back to sleep.
One after another the rest of the sailors do the same: they each take one
fifth of the coconuts in the pile (after giving the extra one to the monkey)
and then return to sleep.
When the sailors awaren the next morning they all notice the coconut pile
is much smaller than it was the night before, but since each man is as guilty
as the others, no one says anything. They divide the coconuts (for the sixth
time) and again there is one left for the monkey.

The question is: How many coconuts were in the original pile?