A very big building in which thousand monkeys are living is lighted by thousand lamps. Every lamp is connected to a unique on/off switch, which are numbered from 1 to 1000. At some moment, all lamps are switched off. But because it is becoming darker, the monkeys would like to switch on the lights. They will do this in the following way.

Monkey 1 presses all switches that are a multiple of 1.
Monkey 2 presses all switches that are a multiple of 2.
Monkey 3 presses all switches that are a multiple of 3.
Monkey 4 presses all switches that are a multiple of 4.
Etc., etc.

How many lamps are switched on after monkey 1000 pressed his switches? And which lamps are switched on?

A spy satellite circles around the earth in 1.57 hours. Can you tell within 3 seconds how much time it takes until the satellite has made 24 revolutions?

How many different squares are located on a chess-board?