Using the distinct digits $a, b$ and $c$, Araceli wrote the number $a b c$, and Luana wrote the numbers $a b, b c$ and $c a$. Find the digits $a, b$ and $c$, knowing that the sum of the numbers written by Luana coincides with the number written by Araceli.

#
Solution

We want the following identity to be satisfied:

$$
a b c=a b+b c+c a
$$

The left side of the equality can be written as $100 a+10 b+c$, while the right side can be written as

$$
(10 a+b)+(10 b+c)+(10 c+a)=11 a+11 b+11 c
$$

Substituting these expressions into the identity $(\uparrow)$, we obtain the equation

$$
89 a=b+10 c
$$

The right side of the last equation is the number $c b$. For $89 a$ to be a two-digit number, necessarily $a=1$. Therefore, $c b=89 a=89$ and, finally, $b=9$ and $c=8$.