Hectictar helped me one of these questions before.
The faces of a standard die are numbered 1, 2, 3, 4, 5, and 6 such that the sum of the numbers on any two opposite faces is 7. Tia writes one number on each vertex of the die such that the number on each face of the die is the greatest common divisor of the numbers at the four vertices of that face. What is the smallest possible sum of the eight numbers Tia writes?
When I started drawing this problem out, I was really focusing on both the aspects of "smallest possible sum" and "greatest common divisor" which for some reason combined in my head to think of using the least common multiple of the 3 sides that make the vertex be the number at the vertex. I think this would also create the smallest possible sum. I'm not entirely sure, though, lmao.
Here is my solution.
Please excuse my poor handwriting, bad lighting, and dirty whiteboard hehehe
Also this is where I do all my math if anyone is curious!!
To clarify, the drawing has visible aspects of the die in purple and hidden aspects of the die in orange. So each vertex becomes the least common multiple of each side that it touches. For example, \(a = lcm(1,3,5) = 15\).
So, my answer is 153. I'm not sure if this is THE lowest answer, but this is what I came up with.