Question 718732: Following recess, the 100 students at a school lined up for the following activity: The first student opened all the 1000 lockers in the school. The second student cloased all lockers with even numbers. The third student "changed" all lockers that were numbered with multiples of 3 by closing those that were open and opening those that were closed. The fourth student changed each locker whose number was a multiple of 4 and so on. After all 1000 students had completed the activity, which lockers were open? Why?
Answer by jkdamm(1)   (Show Source):
You can put this solution on YOUR website! Number the lockers from 1 to 1000.
The problem has to do with the number of divisors of a locker number.
For example; locker number 12.
The divisors of 12 are: 1, 2, 3, 4, 6, and 12.
The locker was opened by the 1st student,
. . then closed by the 2nd student,
. . then opened by the 3rd student,
. . then closed by the 4th student,
. . then opened by the 6th student,
. . then closed by the 12th student.
We can make a general rule.
If the locker number has an even number of divisors,
. . the locker will be closed.
Do any numbers have an odd number of divisors?
We find (perhaps by accident) that squares have an odd number of divisors.
For example: 16 has five divisors
. . . . . . . . . .25 has three divisors
Therefore, the lockers that are left open are:
. .
1, 4 , 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961.
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