In the game of 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 unknown, 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.