Subject 3. Find the natural numbers of the form $\overline{a b c}$ knowing that when divided by $(\overline{a b}+\overline{a c})$ they give a quotient of 5 and a remainder of 5.
## Subject 3.

|  | $\overline{a b c}=(\overline{a b}+\overline{a c}) \cdot 5+5$ |  |
| :--- | :--- | :--- |
|  | $100 a+10 b+c=100 a+5 b+5 c+5$ |  |
|  | $5 b=4 c+5$ so $c=0$ or $c=5$ |  |
|  | If $c=0$ then $b=1$ the numbers are of the form $\overline{a 10}$ where $a$ is a non-zero digit | $1 \mathbf{p}$ |
|  | If $c=5$ then $b=5$ the numbers are of the form $\overline{a 55}$ where $a$ is a non-zero digit | $1 p$ |
|  | The numbers are $110,210, \ldots, 910,155,255, \ldots, 955$ |  |