We built a cuboid from small cubes with an edge of $1 \mathrm{~cm}$. If we removed one column from the cuboid, the remaining structure would consist of 602 cubes. If we removed one row from the top layer instead, the remaining structure would consist of 605 cubes. What are the dimensions of the cuboid?
The edges of the base of the cuboid measure $x \, \text{cm}$ and $y \, \text{cm}$, and the height of the cuboid is $v \, \text{cm}$. If we remove a column, i.e., $v$ cubes, the number of remaining cubes will be divisible by the number $v$. If, instead, we remove a row, i.e., $x$ cubes, the number of remaining cubes will be divisible by the number $x$. Thus, the number 602 is divisible by $v$, and the number 605 is divisible by $x$. The number $602 = 2 \cdot 7 \cdot 43$ has the following divisors (one of which is $v$):

$$
1, 2, 7, 14, 43, 86, 301, 602 \text{.}
$$

The number $605 = 5 \cdot 11 \cdot 11$ has the following divisors (one of which is $x$):

$$
1, 5, 11, 55, 121, 605 \text{.}
$$

In the problem, it is stated that if we remove $v$ cubes, there will be three fewer cubes left than if we remove $x$ cubes. Thus, $x + 3 = v$. Add the number three to each number in row (2) and try to find the resulting number in row (1). This will only succeed in the case of $11 + 3 = 14$. Therefore, the height $v = 14 \, \text{cm}$, and the edge $x = 11 \, \text{cm}$. The total number of cubes is $602 + 14 = 616$. The edge $y = 616 \div 14 \div 11 = 4 \, \text{cm}$.

$[x + 3 = v \ldots 1$ point,

602 is divisible by $v$, 605 is divisible by $x \ldots 2$ points,

$14 \, \text{cm}, 11 \, \text{cm}, 4 \, \text{cm}$ with explanation ... 3 points]