The correct answer is A. tie up.
To tie up means to prevent someone or something from moving or acting freely. In this case, the battalion operating from the mountain was able to prevent three enemy divisions from moving or acting freely by attacking them and keeping them occupied.
The other options are incorrect because they do not accurately describe the situation. Option B, tie down, means to hold someone or something down physically. Option C, tie on, means to attach something to something else using a rope or other material. Option D, tie with, means to be equal to someone or something in a competition or contest.