Imagine there are 1000 lockers and 1000 students at school. The 1st student who comes through the school opens every locker as they enter the school. The 2nd student walks in and closes every other locker begining with locker number two. The 3rd student walks in and changes the state of every third locker begining with locker number three. The next student walks through and changes the state of every 4th locker begining with locker number four. Imagine this continues until all the students have followed the pattern with the 1000 lockers. At the end, which lockers will be open and which lockers will be closed ? Why?

Answer 1

Take fifty cards, labeling each 1,2,3,...50.

Then follow the rules above, removing ards. See what happens.

Answer 2

all the squares since they have an odd number of factors