Computation Workshop Solution Checker

GCD Acrobatics

Let \(n\) be a given positive integer. Consider a set \(X\) of \(n\) different positive integers such that the greatest common divisor of any \(4\) different members of \(X\) is \(1\) however the greatest common divisor of any \(3\) members of \(X\) is more than \(1\). Let \(S\) be the sum of all the members of \(X\). For example if \(n=3\) then the minimum possible value of \(S\) would be \(12\) (which can be achieved by the set \(X = \{ 2,4,6 \}\)).