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?
Certainly, the numbers can all be the same:1,1,1,1,1,1,1
There can be two distinct numbers: 1,1,1,1,1,1,5 (or multiply all by any n)
I cannot find a set with 3 distinct numbers. It seems there will alway be too many different sums too close together for the largest number to divide all of them.
Consider a,...,b,...,c with some number of a's and b's and a largest c.
The sets that don't contain c will have something like 3a+2b and 2a+3b. But c can't divide both sums because they only differ by ab.
Adding a second c doesn't help.

Posted by Jer
on 20190423 16:00:59 