Question 122
There are $n$ lamps, each with a switch. Each lamp works normally: pressing its switch turns it off if it is on, and turns it on if it is off. Every minute we touch the switches.
In the first minute: we press all the switches $(1,2,3, \ldots, n)$.
In the second minute: we press all the switches whose lamp has an even number $(2,4,6, \ldots)$.
In the third minute: we press all the switches whose lamp has a number divisible by $3\ (3,6,9, \ldots)$.
And so on: in the $k$-th minute, we press the switches whose number is divisible by $k$.
Which lamps are on after $n$ minutes, given that at the beginning all the lamps are off?
In the first minute: we press all the switches $(1,2,3, \ldots, n)$.
In the second minute: we press all the switches whose lamp has an even number $(2,4,6, \ldots)$.
In the third minute: we press all the switches whose lamp has a number divisible by $3\ (3,6,9, \ldots)$.
And so on: in the $k$-th minute, we press the switches whose number is divisible by $k$.
Which lamps are on after $n$ minutes, given that at the beginning all the lamps are off?