Problem 5. Martha is playing a computer game called "Destroy the Balloons". In each move, Martha can destroy exactly 1, 11, 15, or 27 balloons. At the same time, if she destroys 1 balloon, exactly 10 new balloons appear on the screen; if she destroys 11 balloons, exactly 8 new balloons appear; if she destroys 15 balloons, no new balloons appear; and if she destroys 27 balloons, exactly 36 new balloons appear. Martha wins if she destroys all the balloons on the screen. If there were 102 balloons on the screen at the beginning, can Martha win?
Solution. Martha can win, as one sequence in which she can destroy the balloons is

$$
15,15,15,15,15,15,11,1,11,15
$$

Note. Let's note that in each step, the difference between the number of balloons that appear and those that are destroyed is a number divisible by 3. The number 102 is divisible by 3. It is not difficult to check that if the number of balloons is a number not divisible by 3, Martha cannot destroy all the balloons in any number of steps.