Factors of successors of powers

Question

Given some $a < 100$ and $b < 10$, is there a fast way to find the number of factors of $(a^b)+1$? What about $(a^b)+2$ or $(a^b)+3$? If this isn't possible, are there any special cases that are? I'm playing a math game and my opponents found a strategy that forces me to solve this problem 1/3 of the time.