4. Determine all triples $(x, y, z)$ of natural numbers such that

$$
x y z + x y + y z + z x + x + y + z = 243
$$
Solution. We add +1 to both sides of the given equation and get

$$
x y z + x y + y z + z x + x + y + z + 1 = 244
$$

which is

$$
x y(z + 1) + x(z + 1) + y(z + 1) + z + 1 = 244
$$

i.e.

$$
(z + 1)(x y + x + y + 1) = 244
$$

from which

$$
(x + 1)(y + 1)(z + 1) = 244
$$

Now from $244 = 2 \cdot 2 \cdot 61$ we get that $(x + 1, y + 1, z + 1)$ is one of the triples $(2,2,61),(2,61,2),(61,2,2)$. Therefore, the desired triples are $(1,1,60)$, $(1,60,1),(60,1,1)$.