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Math Help - Interesting set/divisibility/counting question

  1. #1
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    Interesting set/divisibility/counting question

    A set S, with only a finite number of integers as members, has the property that the average of any 3 of its members is not an integer. What is the maximum number of elements that S may have?
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    Hello, Chris11!

    There is no formula for this problem (that I know of).
    I had to reason it out . . .


    A set S with only a finite number of integers as members,
    has the property that the average of any 3 of its members is not an integer.
    What is the maximum number of elements that S may have?

    There are three types of integers:

    . . (1) A multiple of 3: . 3a
    . . (2) One more than a multiple of 3: . 3b+1
    . . (3) One less than a multiple of 3: . 3c-1


    We must not have 3 of type (1), or 3 of type (2), or 3 of type (3).
    . . Their averages would be integers.

    We can have at most two of one type.


    We must not have one of each type.
    . . Their average would be an integer.

    We can have at most two types of numbers.


    Hence, we can have (at most) two each of two types of numbers.

    Therefore, set S can have a maximum of 4 elements.


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  3. #3
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    I believe the answer is 4.

    If you look at the integers modulo 3, you have three possibilities: 0, 1, and -1. The set S cannot contain one of the following:

    1) 0, 1, and -1
    2) At least three repeating elements.

    If the order of S is at least five, it's going to have one of the two.
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    Yep, that's exactly what I did, Black. Soroban, your solution is interesting in that it introduces notation that I didn't think of. Now try it in the general case!
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    Quote Originally Posted by Chris11 View Post
    Yep, that's exactly what I did, Black. Soroban, your solution is interesting in that it introduces notation that I didn't think of.
    Now try it in the general case!
    If the equivalence classes modulo three are [0],~[1],~\&~[2] note that one of the three must empty. WHY?
    None of the three can have three members.

    What do you mean by the general case?
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    Let S be a subset of the integers with the property that the average of any n of its members is not an integer. What is the maximum number of elements that S may have?
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