## Task 25/74

The dimensions of all cylinders are to be specified in which the measurement numbers of radius and height are natural numbers and in which the surface area and volume are numerically equal.
If the measures of radius and height are denoted by $r$ and $h$ respectively, where $r, h > 0, r, h \in \mathbb{N}$, the task requires the validity of the equation

$$
\pi r^{2} h = 2 \pi r(r + h)
$$

From this, it immediately follows that

$$
r h = 2 r + 2 h \rightarrow r = \frac{2 h}{h - 2} = 2 + \frac{4}{h - 2}
$$

For $r$ to be a natural number, it is necessary and sufficient that $h$ takes one of the values 6, 4, or 3. Thus, there are three cylinders of the required type:

a) $h_{1} = 6; \quad r_{1} = 3; \quad V_{1} = 54 \pi = O_{1}$

b) $h_{2} = 4; \quad r_{2} = 4; \quad V_{2} = 64 \pi = O_{2}$

c) $h_{3} = 3; \quad r_{3} = 6; \quad V_{3} = 108 \pi = O_{3}$