In the operation below, the letters $a, b$, and $c$ are distinct digits and different from 1.

$$
\begin{array}{r}
a b b \\
\times \quad c \\
\hline b c b 1
\end{array}
$$

Determine the values of $a, b$, and $c$.
If the product of $b$ by $c$ ends in 1, then $b \times c$ can be 21 or 81, which means $b \times c=3 \times 7$ or $9 \times 9$. The only possibility to write the product of two distinct numbers less than 10 is $21=3 \times 7$. Thus, we have two possible cases:

1st case: $b=3$ and $c=7$:

$$
\begin{array}{r}
a 33 \\
\times \quad 7 \\
\hline 3731
\end{array}
$$

In this case $\frac{3731}{7}=533$ and, consequently, $a=5$.

2nd case: $b=7$ and $c=3$:

In this case $\frac{7371}{3}=2457$. Therefore, this case does not occur.