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?
Let L be the largest number in the set. And let the other numbers be A,B,C,D,E,F. Without loss of generality let A>=F.
Then by the rule we have L divides A+B+C+D+E and L divides B+C+D+E+F. But then L must also divide the difference (A+B+C+D+E)-(B+C+D+E+F) = A-F.
But since all the numbers are strictly positive we must also have A-F<L. For L to divide A-F and obey the inequality A-F<L it must be the case A-F=0, that is A=F.
A and F are arbitrary members of the subset {A,B,C,D,E,F} so this argument is applicable to every pair of numbers in the subset.
Then all members of the set of seven integers, except for the largest L, must be equal.
So there are then at most two distinct numbers in the set of seven. Jer shows by construction that two distinct values are indeed possible with the set {1,1,1,1,1,1,5}.
Solution
