In a game of Incan basketball, A points are given for a free throw and B points are given for a field goal, where A and B are positive integers. If A=2 and B=5, then it is not possible for a team to score exactly 1 point. Nor is it possible to score exactly 3 points. Are there any other unattainable scores? How many unattainable scores are there if A=3 and B=5? Is it true for any choice of A and B that there are only finitely many unattainable scores? Suppose A and B are known, but it is known that neither A nor B is equal to 2 and that there are exactly 65 unattainable scores. Can you determine A and B? Explain.

Question

In a game of Incan basketball, A points are given for a free throw and B points are given for a field goal, where A and B are positive integers.  If A=2 and B=5, then it is not possible for a team to score exactly 1 point.  Nor is it possible  to score exactly 3 points.  Are there any other unattainable scores?   How many unattainable scores are there if A=3 and B=5?  Is it true for any choice of A and B that there are only finitely many unattainable scores?  Suppose A and B are known, but it is known that neither A nor B is equal to 2 and that there are exactly 65 unattainable scores.  Can you determine A and B?  Explain.