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Math Help - game points

  1. #1
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    game points

    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.
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  2. #2
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    Quote Originally Posted by Fayyaz View Post
    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.
    With m two point shots and n five point shots, you score 2m+ 5n points. You can get X points if there are integers such that 2m+ 5n= X.

    2 divides into 5 twice with remainder 1. That is 5- 2(2)= 1 (bet you knew that!) For X a positive integer, multiply both sides of the equation by X: (X)(5)- (2X)2= (X)(5)+ (-2X)(2)= X. So one solution is m= X, n= -2X or "X five point shots and -2X two point shots give a total of X points". Of course, that is NOT a valid solution because you can't have "-2X" shots for X a positive number.

    However, m= X- 2k, n= -2X+ 5k, for any integer k, also give 5(X- 2k)+ 2(-2X+ 5k)= 5X- 10k - 4X+ 10k= X since the two 10k parts cancel.

    Now, we want to have X- 2k\ge 0 so X/2\ge 2 and -2X+ 5k\ge 0 so k\ge (2/5)X. That is, k must be a an integer between 2X/5 and X/2.

    Certainly there will be such an integer if the diference between those numbers is greater than or equal to 1: X/2- 2X/5\ge 1 or 5X- 4X= X\ge 10. For X= 10, for example, X/2= 5 and 2X/5= 4 so 10 points can be got with 10- 2(4)= 2 5 point shots and 0 2 points shots or with -2(10)+ 5(5)= 5 2 point shots. For any X larger than or equal to 10, we can make X points with 2 and 5 point shots. For X< 10, we have to look at the specific numbers: 9= 5+ 2(2), 8= 4(2), 7= 5+ 2, 6= 3(2), 5= 1(5), 4=2(2) so that, 1 and 3 are the only "unattainable" values.

    Now, what about 3 and 5 point shots? (I think we just recently had a problem about 3 and 5 cent stamps!)

    We can make X shots if there exist integers m and n such that 3m+ 5n= X. 3 divides into 5 once with remainder 2 and 2 divides into 3 once with remainder 1: 3- 2= 1 and then 3- (5- 3)= 2(3)- 5= 1. Multiplying by X, (2X)(3)- X(5)= (2X)(3)+ (-X)(5)= X. Again, -X is not a valid answer, but, again, m= 2X- 5k, n= -X+ 3k give 3(2X- 5k)+ 5(-X+ 3k)= 6X- 15k- 5X+ 15k= X.

    To have non-negative solutions we must have 2X- 5k\ge 0 so 2X/5\ge k and -X+ 3k\ge 0 so k\ge X/3. That is, k must be an integer between 2X/5 and X/3. Such an integer will certainly exist if the distance between X/3 and 2X/5 is greater than or equal to 1: 2X/5- X/3= X/15 \ge 1 or X\ge 15.

    Of course, X= 15= 3(5) or 5(3) and now we know any number larger than 15 can be written as a combination of 3 and 5. What about numbers less than 15? The difference between X/3 and 2X/5 will be less than 1 but there still might be an integer between them. We need to look at the numbers individually. 14= 5+ 9= 1(5)+ 3(3). 13= 10+ 3= 2(5)+ 3. 12= 4(3). 11= 5+ 2(3). 10= 2(5). 9= 3(3). 8= 5+ 3. 7 is not a multiple of 5 or 3 and 7= 5+ 2 so 7 cannot be written as a combination of 3 and 5. 7 points cannot be scored with 3 and 5 point shots.
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  3. #3
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    "We can make X shots if there exist integers m and n such that 3m+ 5n= X. 3 divides into 5 once with remainder 2 and 2 divides into 3 once with remainder 1: 3- 2= 1 and then 3- (5- 3)= 2(3)- 5= 1."

    Hi thanks for your reply. I didnt understand the bit i pasted above with how you use the remainders to derive a new equation. Also, how will i do the last part of the question where both a and b are unknown?
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