At the end of her experiment, are the following lockers open or closed: locker 5, locker 12, locker 100 and locker 856?

Question

The lockers at Lakendra's school are numbered from 1 to 1000. Lackendra did a math experiment at her school. All of the lockers were closed. She opened every second locker. Next, starting over, she went to every third locker: if it was closed, she opened it, and if it was open, she closed it. She continued this pattern; going to every 4th, 5th, 6th....100th locker. At the end of her experiment are the following lockers open or closed: 5, 12, 100, and 856?

Answer 1

if we look at it like this, if it's touched an odd number of times, it's opened. If it's an even numbered, it's closed.

The fifth locker would only be affected by the 1st and 5th experiment, so two times. Two is even, so locker five is closed.

 

12 is affected by:

1, 2, 3, 4, 6, & 12. so six times. Six is even, so closed.

 

100

1, 2, 4, 5, 10, 20, 25, 50, 100, so nine times. Open.

 

856

1, 2, (not 3, so ignore any multiple of that, )4, (no 5, so ignore multiples of that, same with 7) and 8. 856 is a multiplication of 8 and the prime number 107, so that's as high as it will go, since the experiment stops at the 100th pass.

 

1, 2, 4, 8. Four is even, so closed.