Hello all,
Does anyone know of a set of 4 integers S=[a,b,c,d] where where a,b,c,d >0, and that 3 of the integers in S have a common divisor 'x' > 1 , but also that the GCD(a,b,c,d)=1 ?
This is essentially plugging and playing in my mind, but I'm sure there is a more sophisticated method to this madness.
The pattern is this. Suppose you want to find a set of n positive integers for which every subset containing n–1 of them has a common divisor > 1, but the GCD of the whole set of n integers is equal to 1. The method is to take n distinct prime numbers . For , let be the product of all the s except for . Then will divide all the s except for . But there will be no common divisor (greater than 1) of all the s.