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.
How high can the score in Incan basketball be?
Consider prime numbers and other numbers that do not factor into either 2 or 5 (e.g., 9) or 3 and 5.
http://primes.utm.edu/lists/small/1000.txt
I hope this helps.
Well, even though I have no experience in this kind of math, i'd say it is not about factoring into prime numbers. To be honest that was actually the first thing I'v though as well. Besides, what is Incan basketball actually?
I suppose it is about numbers written in the form:
n=a*2+b*5, a,b a positive integer or zero.
It can be proven that when there exist i,j for two non negative integers k,l with k > l so that i*k-j*l=1 that only a finite number of natural numbers cannot be written in the form
n=a*k+b*l, a,b positive integers or zero, where at least one of a,b mustn't be zero.
Seriously man, don't try to cheat if you this is for the PROMYS application.
To determine the unattainable scores in a game of Incan basketball, we can apply a mathematical concept called the Frobenius coin problem or the coin exchange problem.
In general, given two positive integers A and B, the Frobenius coin problem asks for the largest integer that cannot be expressed as a linear combination of A and B with non-negative integer coefficients. In other words, it seeks the largest unattainable score in terms of the given values of A and B.
To find the unattainable scores, we need to use a theorem known as Sylvester's coin problem theorem, which states that for any pair of positive coprime integers A and B, the largest unattainable score is exactly (A - 1) * (B - 1) - 1.
Let's apply this theorem to the given scenarios.
Scenario 1:
A = 2 and B = 5
The largest unattainable score is (2 - 1) * (5 - 1) - 1 = 1 * 4 - 1 = 3.
So, we have determined that the unattainable scores in this scenario are 1 and 3.
Scenario 2:
A = 3 and B = 5
The largest unattainable score is (3 - 1) * (5 - 1) - 1 = 2 * 4 - 1 = 7.
So, we have determined that the unattainable score in this scenario is 7.
Now, let's address the question about the number of unattainable scores.
For any choice of positive integers A and B, there are only finitely many unattainable scores. This is because Sylvester's coin problem theorem provides a formula to determine the largest unattainable score, which means all smaller numbers can be achieved through various combinations of A and B.
Finally, let's consider the last scenario where it is known that neither A nor B is equal to 2, and there are exactly 65 unattainable scores.
According to Sylvester's coin problem theorem, the largest unattainable score is (A - 1) * (B - 1) - 1. We are given that there are exactly 65 unattainable scores, which means the largest unattainable score is 65. So, we have the equation:
(A - 1) * (B - 1) - 1 = 65
Simplifying this equation, we get:
(A - 1) * (B - 1) = 66
Now, we need to find the pair of positive integers A and B that satisfies this equation. We can list down the factor pairs of 66 and check which pair satisfies the mentioned conditions.
The factor pairs of 66 are:
(1, 66), (2, 33), (3, 22), (6, 11)
To satisfy the given condition that neither A nor B is equal to 2, we can rule out the pair (2, 33).
So, the only valid pair is (6, 11), which means A = 6 and B = 11.
Therefore, if it is known that there are exactly 65 unattainable scores and neither A nor B is 2, then A = 6 and B = 11.
By explaining the principles and concepts involved, it becomes clear how to approach and solve these types of problems systematically.