Bob's Buttons Investigation Guide 2

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  1. Bob found that he could make up problems about his pile of buttons that had no solutions, eg:
    In groups of 3, I have 2 over, and in groups of 6, I have 1 over. What is the smallest number of buttons I could have?
    1. Is there a solution for Bob's new problem? Why or why not?
    2. Make up at least three more problems like Bob's 'trick' problem.
    3. What's different about these problems that causes no solutions?
    4.  

    5. Here is a special kind of buttons problem. It is special because the sizes of the groups are consecutive numbers, and because one group size has no buttons over.
      In groups of 4 there are none over, but in groups of 3 there are 2 over. What is the smallest number of buttons I could have?
      1. Solve this problem.
      2. Make up another problem like it, and solve it too.
      3. Find a general method for solving this kind of problem.
    6.  

    7. Here is a different kind of buttons problem. This time the group sizes have a common factor, and one group size has no buttons over.
      In groups of 8 there are none over, but in groups of 6 there are 4 over. What is the smallest number of buttons I could have?
      1. Solve this problem. Also solve: groups of 4 with none over, and groups of 3 with 2 over. How are these two problems alike?
      2. Make up another problem like it, and solve it too.
      3. Find a general method for solving this kind of problem.
    8.  

    9. Here are some examples of a different kind of buttons problem. It is different because the sizes of the groups:
      • have no common factor,
      • are not consecutive,
      • and because one group size has no buttons over.
      Solve each problem to find the smallest number of buttons.
      1. Groups of 7 with none over, and groups of 5 with 1 over.
      2. Groups of 7 with none over, and groups of 5 with 2 over.
      3. Groups of 7 with none over, and groups of 5 with 3 over.
      4. Groups of 7 with none over, and groups of 5 with 4 over.
      5. Find a general method for solving this kind of problem.
    10.  

    11. All the problem types above had no remainder for one group size. What happens to the solutions if this is not the case?

     

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