A hotel has 100 rooms, numbered 1-100, and each room has a blue door. The manager decides the doors need new paint, so she has them all painted red.
Then the manager decides that she would like every other door painted blue, beginning with door number 2. So the painter dutifully paints every other door blue.
Next, the manager decides that beginning with door 3, to mix things up, she wants the painter to change the current color of every 3rddoor; so if the door is red, paint it blue, and if it’s blue, paint it red. So the painter orders more paint and gets to work.
What next? You guessed it! The manager wants every 4thdoor to have its color changed, beginning with door number 4. Then she wants every 5thdoor’s color changed, beginning with door number 5, then every 6thdoor changed, … and so on to the 100thdoor!
a. When the painter has finally finished (assuming he didn’t quit!), which doors are painted red?
b. Which doors have had their color changed exactly twice?
c. Which door(s) was (were) painted the greatest number of times?
