863. The product of three prime numbers is 5 times their sum. Find all such triples of numbers.
$\triangle$ Let's denote these prime numbers as $x, y$, and $z$. Then

$$
x y z=5(x+y+z)
$$

One of the factors on the left side of the equation is equal to 5, for example, $x=5$:

$$
5 y z=5(5+y+z), \quad y z=y+z+5
$$

Express $z$ in terms of $y$ from the last equation and separate the integer part of the resulting fraction:

$$
y z-z=y+5, \quad z=\frac{y+5}{y-1}=\frac{(y-1)+6}{y-1}=1+\frac{6}{y-1}
$$

From this, $(y-1) \in\{1 ; 2 ; 3 ; 6\}$, and then $y \in\{2 ; 3 ; 4 ; 7\}$. The value $y=4$ should be discarded. In the other three cases, find the corresponding values of $z: z=7,4,3,2$. Since the value $z=4$ is not suitable, the value $y=3$ should also be discarded.

Answer: $2,5,7$.