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.
For a=2 and b=5 i think there isn't anymore unattainable scores. Having a=2 you can get all even scores (2, 4, 6, 8, ....) and i think you get all odd scores except 1 and 3 by combining 2 and 5.
For a=3 and b=5 the only scores you can't get is 1, 2, 4, 7 (not 100% sure again).
Now i will try to take a shot how you can figure it out...
Let x be the unattainable scores, then x= b+a / b-a (rounding x if necessary)
NOT SURE FOR THESE ABOVE, TAKE ANOTHER OPINION.