4. Determine the number of the form $\overline{a b c}$ such that $10 \cdot\left(\frac{\overline{a b}}{c}-1\right)+\frac{\overline{b c}}{a}=82$.

Nicolae Stănică, G.M.
## Solution.

$$
\begin{aligned}
& \frac{10 \cdot \overline{a b}}{c}+\frac{\overline{b c}}{a}=92 \text { or } \frac{\overline{a b 0}}{c}+\frac{\overline{b c}}{a}=92 \text { or } \frac{\overline{a b c}-c}{c}+\frac{\overline{a b c}-100 a}{a}=92 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . .3 \text{ m} \\
& \frac{\overline{a b c}}{c}-1+\frac{\overline{a b c}}{a}-100=92 \text { or } \overline{a b c} \cdot\left(\frac{1}{c}+\frac{1}{a}\right)=193 \text { or } \overline{a b c} \cdot(a+c)=193 \cdot a \cdot c \ldots \ldots \ldots \ldots \ldots . .2 \text{ p}
\end{aligned}
$$

Since 193 is a prime number, we obtain that $\overline{a b c}: 193$ ..... $1 \text{ p}$
Thus $\overline{a b c} \in\{193 ; 386 ; 579 ; 772 ; 965\}$ and only 386 satisfies the relation from the problem statement. ..... $1 \text{ p}$