9.3. Let the two-digit numbers ${ }^{\overline{a b}}$ and ${ }^{\overline{c d}}$ be such that the ratio of the four-digit number $\overline{a b c d}$
to the sum $\overline{a b}+\overline{c d}$ is an integer. Find all possible values that this number can take.
Answer. All natural numbers from 11 to 90 inclusive.

Solution. Let the ratio in question be denoted by ${ }^{n}$, and the numbers $\overline{a b}$ and $\bar{c} \bar{d}$ by $A$ and $C$ respectively. Then $\overline{a b c d}=100 \cdot A+C=n(A+C)$, from which $\frac{100-n}{n-1}=\frac{C}{A}$. Note that $C \leq 99, A \geq 10$, hence $\frac{100-n}{n-1} \leq \frac{99}{10}$, from which $n \geq \frac{1099}{109}$, that is, $n \geq 11$. Similarly, $C \geq 10, A \geq 99$, hence $\frac{100-n}{n-1} \geq \frac{10}{99}$, from which $n \leq \frac{9910}{109}$, that is, $n \leq 90$.

On the other hand, for any natural $n$ in the interval from 11 to 90, setting $C=100-n \cdot A=n-1$, we get two two-digit numbers for which the ratio in the condition will be exactly ${ }^{n}$.

Grading criteria. Estimates $n \geq 11$ and $n \leq 90$: each 2 points. Construction of an example for all natural numbers from 11 to 90: 3 points. If examples are provided not for all natural numbers from 11 to 90, but examples for 11 and 90 are present: 2 points. Any other number of partial examples: 1 point.