Grandma is not yet 100 years old, the granddaughter is more than 10 years old, and Grandma's age is a multiple of the granddaughter's age. When the granddaughter wrote down Grandma's age and then her own, she got a four-digit number. When Grandma wrote down the granddaughter's age and then her own, she got a different four-digit number. The difference between these two four-digit numbers is 7128.

How old can Grandma and the granddaughter be? List all possibilities.

(L. Hozová)
Let $v$ be the age of the granddaughter and $b$ be the age of the grandmother. The age of the grandmother is a multiple of the age of the granddaughter, i.e., $b = k v$ for some natural number $k$.

The four-digit number written by the granddaughter was $100 b + v$, and the four-digit number written by the grandmother was $100 v + b$, so

$$
(100 b + v) - (100 v + b) = 7128 \text{.}
$$

After rearranging (and substituting $b = k v$), we get

$$
\begin{aligned}
99(k v - v) & = 7128 \\
v(k - 1) & = 72.
\end{aligned}
$$

The task is to find $v$ and $k$ such that the above equation holds and additionally $v > 10$ and $b = k v < 100$. We determine the corresponding $k$ and $b = k v$:

| $v$ | 72 | 36 | $\mathbf{24}$ | $\mathbf{18}$ | $\mathbf{12}$ |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $k$ | 2 | 3 | 4 | 5 | 7 |
| $b$ | 144 | 108 | $\mathbf{96}$ | $\mathbf{90}$ | $\mathbf{84}$ |

The strongly marked results are the valid solutions, i.e., those for which $b < 100$. The problem has three solutions.

Another solution. Let $v$ be the age of the granddaughter and $b$ be the age of the grandmother, and let $v = \overline{A B}$ and $b = \overline{C D}$ be the decimal representations of these numbers. The information about the difference of the four-digit numbers from the problem can be expressed using an algebrogram

$$
\begin{array}{r}
C D A B \\
- A B C D \\
\hline 7128
\end{array}
$$

Since the granddaughter is younger than the grandmother, there is a "carry over" on the last two digits, i.e.,

$$
\begin{array}{r}
1 A B \\
- \quad C D \\
\hline 28
\end{array}
$$

Thus, the grandmother is 72 years older than the granddaughter.

Since the granddaughter is at least 11 years old, the grandmother is at least 83 years old. Since the grandmother is at most 99 years old, the granddaughter is at most 27 years old. It is sufficient to check all pairs of $v$ and $b = v + 72$ within these limits and verify whether $b$ is a multiple of $v$:

| $v$ | 11 | $\mathbf{12}$ | 13 | 14 | 15 | 16 | 17 | $\mathbf{18}$ | 19 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| $b$ | 83 | $\mathbf{84}$ | 85 | 86 | 87 | 88 | 89 | $\mathbf{90}$ | 91 |


| $v$ | 20 | 21 | 22 | 23 | $\mathbf{24}$ | 25 | 26 | 27 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| $b$ | 92 | 93 | 94 | 95 | $\mathbf{96}$ | 97 | 98 | 99 |

The valid results are marked in bold; the problem has three solutions.

Evaluation. 1 point for each valid solution; 3 points for completeness and quality of the commentary.