## Task 2 - 200832

Determine all three-digit natural numbers $n$ with the property that the product of the individual digits of $n$ is equal to five times the cross sum of $n$!
If a three-digit natural number $n$ has the required property, then for its three digits, denoted in any order as $a, b, c$,

$$
a b c = 5(a + b + c)
$$

If one of the digits $a, b, c$ were equal to 0, then $a b c = 0$, and according to (1), $a + b + c = 0$. From $a \geq 0, b \geq 0, c \geq 0$, it follows that $a = b = c = 0$, which contradicts the fact that $n$ is a three-digit number. Therefore,

$$
1 \leq a \leq 9, \quad 1 \leq b \leq 9, \quad 1 \leq c \leq 9
$$

According to (1), the prime number 5 is a divisor of $a b c$, and thus of (at least) one of the digits $a, b, c$. Due to (2), this digit is equal to 5; because of the arbitrary choice of order, we can assume $c = 5$. From this, it follows from (1) that

$$
a b = a + b + 5
$$

From (3) it follows that

$$
a b - a - b + 1 = 6 \quad \Rightarrow \quad (a - 1)(b - 1) = 6
$$

According to (2), $a - 1$ and $b - 1$ are also natural numbers; due to the arbitrary choice of order, we can assume $a \leq b$, so there are only two possibilities for (4):

- $a - 1 = 1, b - 1 = 6$, thus $a = 2, b = 7$
- $a - 1 = 2, b - 1 = 3$, thus $a = 3, b = 4$

Considering all possible permutations of $a, b, c$, it follows that only the numbers

$$
257, 275, 527, 572, 725, 752, 345, 354, 435, 453, 534, 543
$$

can have the required properties. They do, as a check shows.