In the 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 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.
To determine if a given score is attainable in the game of Incan basketball, we can make use of the Chicken McNugget theorem.
The Chicken McNugget theorem states that if two positive integers a and b are relatively prime (i.e., their greatest common divisor is 1), then the largest positive integer that cannot be expressed as a combination of a and b (using non-negative integer multiples) is given by ab - a - b.
Let's examine the given cases:
1) When a = 2 and b = 5:
The largest unattainable score is given by 2 * 5 - 2 - 5 = 3. It is confirmed that 1 and 3 are unattainable scores.
2) When a = 3 and b = 5:
The largest unattainable score is given by 3 * 5 - 3 - 5 = 7. With these values, the unattainable scores are 1, 3, 4, 6, and 7. Therefore, there are 5 unattainable scores in this case.
Now, let's consider the general case when a and b are unknown but not equal to 2, and it is known that there are exactly 65 unattainable scores.
If there are exactly 65 unattainable scores, it means that the largest unattainable score is 65.
Using the Chicken McNugget theorem, we can set up the equation ab - a - b = 65.
We know that a and b are positive integers greater than 2. To simplify the equation, let's substitute a = x + 2 and b = y + 2. This substitution allows us to remove the restriction on a and b being greater than 2.
Plugging in these values, the equation becomes:
(x + 2)(y + 2) - (x + 2) - (y + 2) = 65.
Expanding and simplifying, we get:
xy + 2x + 2y - x - y - 2 = 65,
xy + x + y - 2 = 65,
xy + x + y = 67.
At this point, we need to find the possible factor pairs of 67 in order to determine the values of x and y that satisfy the equation. Since 67 is a prime number, its only factor pairs are (1, 67) and (67, 1).
If we solve for x and y separately, we find two possible sets of values:
For x = 1 and y = 66: a = x + 2 = 3 and b = y + 2 = 68.
For x = 66 and y = 1: a = x + 2 = 68 and b = y + 2 = 3.
Therefore, the possible values for a and b are (3, 68) and (68, 3).
In summary, if it is known that neither a nor b is equal to 2 and there are exactly 65 unattainable scores, then the possible values for a and b are either (3, 68) or (68, 3).