## Problem 4

Find the numbers of the form $\overline{a b c}$ knowing that when dividing $\overline{a b c}$ by $\overline{b c}$, the quotient is 6 and the remainder is 5.

Note

- Effective working time 2 hours.
- All problems are mandatory.
- 7 points are awarded for each correctly solved problem.


## National Mathematics Olympiad  Local stage -14.02.2015  Grading criteria Grade 5
## Problem 4

According to the natural number theorem, we have $\overline{a b c}=6 \cdot \overline{b c}+5$ $1 p$

$\overline{b c}=20 a-1$ 2p

For $a=1=>\overline{b c}=19=>\overline{a b c}=119$

For $a=2=>\overline{b c}=39=>\overline{a b c}=239$

For $a=3=>\overline{b c}=59=>\overline{a b c}=359$

For $a=4=>\overline{b c}=79=>\overline{a b c}=479$

For $a=5=>\overline{b c}=99=>\overline{a b c}=599$

For $a \geq 6$ there are no solutions......................................................................................3p

![](https://cdn.mathpix.com/cropped/2024_06_07_dfc9162f0525ae4a7f10g-3.jpg?height=68&width=1545&top_left_y=676&top_left_x=274)