Seven positive integers are written on a piece of paper. No matter which five numbers one chooses, each of the remaining two numbers divides the sum of the five chosen numbers.
How many distinct numbers can there be among the seven?
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If we call the set containing the seven numbers S = ( a , b , c , . . . , g ) then g must divide a + b + c + d + e and a + b + c + d + f . This means that g must also divide the difference namely e − f . This case can of course be generalized to every number such that every number in S divides the difference between two randomly chosen numbers in S . Assume that there exists three distinct numbers x , y , z in S such that x < y < z . Then z must divide y − x but that is impossible since y − x < z and y − x is positive. Therefore there can't be three distinct numbers so there can only be 2 distinct numbers in S at max.
One example is S = 5 , 1 , 1 , 1 , 1 , 1 , 1