253. Find the dimensions of a rectangular parallelepiped if they are expressed as integers, and the total surface area and volume are numerically equal.
253. We obtain the equation $x y z=2(x y+x z+y z)$, or

$$
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{2}
$$

Hint. First find equal values of $x, y, z$, then consider the case of two equal unknowns, finally the general case. As a result, we get 10 solutions: $(6,6,6),(5,5,10),(4,8,8),(3,12,12),(3,7,42),(3$, $8,24),(3,9,18),(3,10,15),(4,5,20),(4,6,12)$.