Pan Vlk was waiting for the bus at a stop in front of the school. From the window, he heard the teacher say:

"How large can the surface area of a regular quadrilateral prism be, if you know that the lengths of all its edges are expressed in whole numbers in centimeters and that its volume is... "

This important number Pan Vlk did not hear because a car just passed by. A little later, he heard a student reporting the result of $918 \mathrm{~cm}^{2}$. The teacher replied:

"Yes, but the problem has four solutions in total. Keep looking."

Pan Vlk did not learn more because he boarded his bus. Since mathematics had always been his hobby, he took out a pencil and paper in the bus and, after some time, determined the remaining three solutions to the teacher's problem. Calculate them as well.

(L. Šimunek)
Variables $a$ and $v$ are natural numbers representing the edge of the base and the height of a regular quadrilateral prism, respectively. For the dimensions considered by the student who signed up, the following equation holds:

$$
918=2 a^{2}+4 a v=2 a \cdot(a+2 v) \text {, }
$$

after dividing by two, we get

$$
459=a \cdot(a+2 v) .
$$

We will look for all pairs $(a, v)$ that satisfy this relationship. Therefore, we will determine all possible factorizations of the number $459\left(459=3^{3} \cdot 17\right)$ into the product of two natural numbers, where the smaller one will be $a$ and the larger one will be $a+2 v$. The following table shows that there are four such factorizations, each leading to an integer $v$. For all found pairs $(a, v)$, we will then calculate the volume that the teacher must have assigned, and its prime factorization:

|  | $a$ | $a+2 v$ | $v$ | $a^{2} \cdot v$ |
| :---: | :---: | :---: | :---: | :---: |
| 1. option | 1 | 459 | 229 | $1^{2} \cdot 229=229$ |
| 2. option | 3 | 153 | 75 | $3^{2} \cdot 75=3^{3} \cdot 5^{2}$ |
| 3. option | 9 | 51 | 21 | $9^{2} \cdot 21=3^{5} \cdot 7$ |
| 4. option | 17 | 27 | 5 | $17^{2} \cdot 5=17^{2} \cdot 5$ |

The teacher revealed that the assigned volume leads to four solutions. For each volume in the table, we will determine how many solutions it leads to, i.e., for each volume, we will find all possible $a$:

|  | $a^{2} \cdot v$ | possible $a$ |
| :---: | :---: | :---: |
| 1. option | 229 | 1 |
| 2. option | $3^{3} \cdot 5^{2}$ | $1,3,5,15$ |
| 3. option | $3^{5} \cdot 7$ | $1,3,9$ |
| 4. option | $17^{2} \cdot 5$ | 1,17 |

We see that only the 2. option leads to four prisms. Therefore, the teacher assigned the volume $3^{3} \cdot 5^{2} = 675 \left(\mathrm{~cm}^{3}\right)$, and the first student considered these dimensions: $a=3 \mathrm{~cm}, v=75 \mathrm{~cm}$. Below, we list the other dimensions of the prism that the students were expected to find and the surface area they were to calculate:

| $a$ | 1 | 5 | 15 |
| :---: | :---: | :---: | :---: |
| $v$ | 675 | 27 | 3 |
| $2 a^{2}+4 a v$ | 2702 | 590 | 630 |

The teacher expected the following additional three solutions: $590 \mathrm{~cm}^{2}, 630 \mathrm{~cm}^{2}$, and $2702 \mathrm{~cm}^{2}$.