A bunch of math teachers decided to create their own football league where they

Question

A bunch of math teachers decided to create their own football league where they changed the rules of the game some what. Each field goal counts 5 points and each touchdown counts for 3 points. Obviously they preferred kicking to running. They did not allow any other forms of scoring points. Since they were more thinkers than runners, one of the team members noticed that some scores were impossible to get. For example you cannot have a score of 1 or 2 or 4. The question that they got distracted by was whether there is a highest number that can NOT be a score in the game. Of course they sit on the bleachers and start to calculate.

1. To get these players back on the field, determine the highest number than cannot be a score and explain why your are sure there are not higher numbers. Post your answer on this form as a challenge to your classmates. In order to challenge you they must find how many 3 point scores and how many 5 point scores are needed to make your number. Look back every so often and see if you have stumped everyone, if so you will get extra points.

2. Try some other scoring patterns and determine if they have scores that are impossible to achieve. Of these are there some scoring patterns that do NOT have a highest impossible score? See if you can find a rule that will let you know if there is or if there is not a highest impossible score.

3. For those scoring patterns where you do have a highest impossible score, see if you can figure out a formula to determine what that highest impossible score is.

4. How is this connected to least common multiples?

Explanation / Answer

Here different scores that we can set will be of form 2+5= 8

3+3+3= 9

8+3= 11

5+5=10

9+3=12

and so on

that means the minimum score that we can make here is 8.

That means a score of 7 can never be touched. So the highest required number that can not be scored is 7.

This is the answer of question (1).

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