2. Determine the numbers of the form $\overline{abc}$ which, when divided by $\overline{bc}$, give a quotient and a remainder both equal to 18.

Daniela Cerchez, Brăila
## Solution.

![](https://cdn.mathpix.com/cropped/2024_06_07_237c5a93d605dcd18d26g-2.jpg?height=74&width=1534&top_left_y=1639&top_left_x=244)

![](https://cdn.mathpix.com/cropped/2024_06_07_237c5a93d605dcd18d26g-2.jpg?height=86&width=1534&top_left_y=1759&top_left_x=244)
We have $17b + 12 = 10a \Rightarrow U(17b + 12) = U(10a) = 0 \Rightarrow b = 4 \Rightarrow a = 8$ ..... $3 \text{p}$
In conclusion, $\overline{abc} = 846$ ..... $1 \text{p}$