|Spoiler Inside: Solution to Puzzle||SelectShow>|
No for all cases.
The problem starts with a positive integer. Each step, regardless of which part of the step is done, gives an integer result that is bigger than at the beginning of the step. The last step will then start with a number that is an integer greater than 1 which will then be multiplied by either 2 or 3. The result can not be prime as it has factors of the starting number for the last step and either 2 or 3.