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 2nd door painted blue, beginning with door number 2. So the painter dutifully paints every 2nd 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 3rd door; 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 4th door to have its color changed, beginning with door number 4. Then she wants every 5th door’s color changed, beginning with door number 5, then every 6th door changed, … and so on to the 100th door!

a. When the painter has finally finished (assuming he didn’t quit!), which doors are painted red? List the door numbers and justify your work with a written explanation.

b. Which doors have had their color changed exactly twice? List the door numbers and justify your work with a written explanation.

c. Which door(s) was (were) painted the greatest number of times? Justify your work with a written explanation.