2005 AIME II Problems/Problem 5
Determine the number of ordered pairs of integers such that and
This problem needs a solution. If you have a solution for it, please help us out by.
Note that the equation can be rewritten as: or Multiplying through by and factoring yields . Therefore, or . Therefore, either or . For the case , note that and . Thus, all values of a from 2 to 44 will work. For the case , note that while . Therefore, for this case, all values of a from 2 to 12 work. There are possibilities for the square case and possibilities for the cube case. Thus, the answer is .