## Task B-2.4.

The lengths of the edges of a rectangular prism are natural numbers. When we add the volume, half of the surface area, and the lengths of the edges, we get 1770. Determine the lengths of the edges of the rectangular prism.
## Solution.

Let $a, b, c$ be the lengths of the edges of the cuboid. Without loss of generality, let $a \leqslant b \leqslant c$. Then, from the condition in the problem, we have:

$$
a b c + a b + a c + b c + a + b + c = 1770
$$

If we add 1 to both sides of the equation, we can group the terms of the given expression and write the left side in the form of a product.

$$
\begin{aligned}
& (a b c + a b) + (a c + a) + (b c + b) + c + 1 = 1771 \\
& a b (c + 1) + a (c + 1) + b (c + 1) + c + 1 = 1771 \\
& (c + 1)(a b + a + b + 1) = 1771 \\
& (c + 1)(a (b + 1) + b + 1) = 1771 \\
& (a + 1)(b + 1)(c + 1) = 7 \cdot 11 \cdot 23
\end{aligned}
$$

The expression on the left side is a product of natural numbers, and the right side is the prime factorization of 1771. Therefore, the only possibility is

$$
a + 1 = 7, b + 1 = 11, c + 1 = 23
$$

which gives

$$
a = 6, b = 10, c = 22
$$