Question 18
On each face of a cube, an integer greater than $2$ is written. Each vertex of the cube is the intersection of three unique faces, and each edge is the intersection of two unique faces. Assign to each vertex the product of the numbers written on the faces intersecting the vertex, and assign to each edge the product of the numbers written on the faces intersecting the edge. The sum of the numbers assigned to the eight vertices is equal to $2024$. Find the maximum possible value of an edge.