Interesting set/divisibility/counting question

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The discussion centers on a mathematical problem regarding a set S of integers where the average of any three members is not an integer. The conclusion is that the maximum number of elements in set S is 4, specifically from two residue classes, which can be exemplified by the integers 1, 2, 4, and 5. The reasoning involves analyzing the conditions under which the sum of any three integers results in a multiple of three, leading to the determination of allowable combinations.

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  • Understanding of integer properties and residue classes
  • Basic knowledge of averages and their calculations
  • Familiarity with modular arithmetic
  • Concept of combinatorial counting in set theory
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This discussion is beneficial for mathematicians, educators, and students interested in number theory, particularly those exploring properties of integers and combinatorial mathematics.

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Suppose that a set S has the property that all of its members are integers and 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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The average of three members a,b,c is (a+b+c)/3 and is an integer iff a+b+c=0(3).

If any three members are such that a=b=c(3) then a+b+c=0(3). Similarly if a,b,c are in 0,1,-1(3) in some order a+b+c=0(3).

Therefore the max is 2 from each of 2 residue classes, which is achievable e.g.

1,2,4,5
 

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