4. On each side of the cube, there is a natural number written. Each vertex of the cube is associated with the product of the three numbers that are on the faces containing that vertex. The sum of all eight numbers associated with the vertices is 455. What is the sum of all the numbers written on the sides of the cube?
4. On each side of the cube, a natural number is written. Each vertex of the cube is associated with the product of the three numbers that are on the faces that determine that vertex. The sum of all eight of these numbers is 455. What is the sum of all the numbers written on the sides of the cube?

## Solution.

Let the vertices of the cube be standardly labeled as $A, B, \ldots, H$.

Let the number written on the bottom face be $x$, and on the top face $a$.

Let the number written on the front face be $y$, and on the back face $b$.

Let the number written on the left face be $z$, and on the right face $c$.
![](https://cdn.mathpix.com/cropped/2024_05_30_ba00de53bdad84e80dffg-17.jpg?height=432&width=1558&top_left_y=2047&top_left_x=250)

Then, the number associated with vertex $A$ is $x y z$, with vertex $B$ is $x y c$, with vertex $C$ is $x b c$, with vertex $D$ is $x z b$, with vertex $E$ is $y z a$, with vertex $F$ is $y a c$, with vertex $G$ is $a b c$, and with vertex $H$ is $z a b$.

We have:

$$
\begin{gathered}
x y z + x y c + x b c + x z b + y z a + y a c + a b c + z a b = 455 \\
x \cdot (y z + y c + b c + z b) + a \cdot (y z + y c + b c + z b) = 455 \\
(y z + y c + b c + z b) \cdot (x + a) = 455 \\
{[y \cdot (z + c) + b \cdot (c + z)] \cdot (x + a) = 455} \\
(z + c) \cdot (y + b) \cdot (x + a) = 455
\end{gathered}
$$

The prime factorization of 455 is:

$$
455 = 5 \cdot 7 \cdot 13
$$

Clearly, the sums of the numbers on opposite faces of the cube are 5, 7, and 13 (in any order).

Then, the sum of all the numbers written on the cube is

$$
x + y + z + a + b + c = (z + c) + (y + b) + (z + c) = 5 + 7 + 13 = 25
$$