A rectangular prism has edges of lengths $a, b, c$ which are integers, and the numerical value of its surface area is equal to that of its volume. What are the lengths of its edges, given that $c=\frac{a \cdot b}{2}$?
The volume of the rectangular prism is $a \cdot b \cdot c$, and its surface area is $2(a b+b c+c a)$. By substituting $c=\frac{a b}{2}$, we arrive at the equation:

$$
a b \cdot \frac{a b}{2}=2\left(a b+\frac{a b^{2}}{2}+\frac{a^{2} b}{2}\right)
$$

Multiplying both sides by $\frac{2}{a b}$:

$$
a b=4+2 b+2 a, \quad (2) \quad a(b-2)=4+2 b \quad a=2+\frac{8}{b-2}
$$

Since $a$ and $b$ are integers, it is necessary that $b-2$ be a divisor of 8.

If $b-2=-8$ or -8, then the dimensions of the rectangular prism are $3,10,15$.

If $b-2=8$, then the dimensions of the rectangular prism are $3,10,15$.

If $b-2=4$, then the dimensions of the rectangular prism are $4, 6, 12$.

If $b-2=2$, then the dimensions of the rectangular prism are $4, 6, 12$.

If $b-2=1$, then the dimensions of the rectangular prism are $3, 10, 15$.

If $b-2=-1$, then $a$ would be negative.

If $b-2=-2$, then $b=0$.

If $b-2=-4$ or -8, then $b$ would be a negative number.

Thus, we get two solutions: the rectangular prisms with edge lengths $3,10,15$ and $4, 6, 12$.

Note. If we are a bit more clever, we can rearrange the above (2) formula symmetrically:

$$
(a-2)(b-2)=8
$$

We need to determine $a$ and $b$ such that they are positive integers, and the product of $a-2$ and $b-2$ is 8. The possibilities are: $1 \cdot 8 ; 2 \cdot 4 ;(-1) \cdot(-8) ;(-2) \cdot(-4)$, which can be examined similarly to the first solution.