TR-MO 2018 Problem 5

Let $a_1,a_2,a_3,a_4$ be positive integers, with the property that it is impossible to assign them around a circle where all the neighbors are coprime. Let $i,j,k\in\{1,2,3,4\}$ with $i \neq j$ , $j\neq k$ , and $k\neq i$ . Determine the maximum number of triples $(i,j,k)$ for which $$({\rm gcd}(a_i,a_j))^2|a_k.$$