634 Perfect numbers
Exercise
1.The isle of Pythagora, while very sparsely populated, is capable
of supporting a population of thirty million. On the 6th day of
the 28th anniversary of his accession to the throne, the king of the
island called a meeting of his 496 advisors to decide how to cel-
ebrate the auspicious occasion. They decided to divide the regal
jewels among the people of the land. All the people, including the
king the advisors, were lined up in a single file, and the jewels were
distributed as follows.
Starting with the second in line, each person was given one jewel.
Starting with the 4th in line, each second person was given two
jewels.
Starting with the 6th in line, each third person was given three jew-
els.
Starting with the 8th in line, each fourth person was given four jew-
els, and so on.
The man at the extreme end of the line noticed that the number of
jewels he received corresponded to his position in line.
How many people were there in Pythagora?2.A minimum security prison contains 100 cells with one prisoner
in each. The athletic young warden was ordered to free a certain
number of these prisoners at his discretion, and this is how he did
it.
First he walked along the row of cells opening every door. Starting
at the beginning again, he shuts every second door. During his
third walk, starting at the beginning, he stopped at every third door:
if it was open he shut it, if it was shut he opened it. On his fourth
walk he did the same, opening closed doors and closing open doors,
except he did it for every fourth door. On his fifth walk he stopped
at every fifth door, closing it if it was open and opening it if it was
shut. And so on, until at last he had completed the full hundred
walks.
The prisoners in cells whose doors were still open were freed.
Which were the lucky cells?3.For a positive integern, show that ifσ(n)is prime, then so isd(n).