## Subject II. (20 points)

Determine the digits $a, b, c$ knowing that dividing 2216 by $\overline{5 a}$ results in the quotient $\overline{4 b}$ and the remainder $\overline{2 c}$, where $\overline{2 c}$ is a prime number.

Prof. Gheorghe Lobont, "Mihai Viteazul" National College, Turda
## Subject II.

From the division theorem with remainder, we obtain $2216=\overline{5 a} \cdot \overline{4 b}+\overline{2 c}$ and from the hypothesis $c \in\{3,9\}$.

If $c=3$ then $u(a \cdot b)=3$ which leads to $a, b \in\{1,3\}$, with the solutions $a=1$ and $b=3$, respectively $a, b \in\{7,9\}$, but in this case, there are no solutions.

(5p)

If $c=9$ then $u(a \cdot b)=7$ which leads to $a, b \in\{1,7\}$, respectively $a, b \in\{3,9\}$ and there are no solutions. (5p)

Therefore, the solution to the problem is $\left\{\begin{array}{l}a=1 \\ b=3 \\ c=3\end{array}\right.$.