A number theory problem by Finn C
There are 100 people in a room. There are also 100 closed lockers. Each of the 100 people are assigned a number from 1-100. The first person with the number 1, would open every locker since every number from 1-100 is a multiple of one. The second person would close every second locker. The third person would open or close every third locker, for example if locker 52 was open, he would close it. If number 52 was closed he would open it. This process is repeated until every person has opened or closed the lockers they're meant to. Which lockers will be open?
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1 solution
Moderator note:
Why is it true that "the only numbers with an odd amount of factors are squared numbers"?
The answer is squared numbers.
This is true because every number, except square numbers have an even number of factors, because they come in pairs such as 6 and 4, and 8 and 3. Given that they start off closed, an even number of factors would take it to be opened, then closed again.
Square numbers are different because they are opened, closed - but since there is an odd number of factors (including themselves) - it would be opened again.
Since the only numbers with an odd amount of factors are squared numbers - squared numbers are the answer