Estonian Mathematical Olympiad

Let $n$ satisfy the condition and $n'$ be the number obtained by the replacement of digits. Denote by $x$ the number formed by the digits to the left of the replaced digit and denote by $y$ the number formed by the digits to the right of the replaced digit. Let $k$ be the number of digits in $y$. Then $n = x \cdot 10^{k+1} + 4 \cdot 10^k + y$ and $n' = x \cdot 10^{k+2} + 22 \cdot 10^k + y$. As $n'$ is divisible by $n$, then so must also be $n' - n$ and $10n - n'$, which yields
$$x \cdot 10^{k+1} + 4 \cdot 10^k + y \mid 9x \cdot 10^{k+1} + 18 \cdot 10^k, \qquad (2)$$
$$x \cdot 10^{k+1} + 4 \cdot 10^k + y \mid 18 \cdot 10^k + 9y. \qquad (3)$$
We will consider the following cases.

* If $x = 0$, then by (2), $t (4 \cdot 10^k + y) = 18 \cdot 10^k$ for some integer $t$. As
$5 (4 \cdot 10^k + y) \ge 5 \cdot 4 \cdot 10^k > 18 \cdot 10^k > 3 \cdot 5 \cdot 10^k > 3 (4 \cdot 10^k + y)$,
the only option is $t = 4$. The equation $4 (4 \cdot 10^k + y) = 18 \cdot 10^k$ yields
$y = \frac{2 \cdot 10^k}{4} = 5 \cdot 10^{k-1} < 10^k$. Altogether $n = 45 \cdot 10^{k-1}$.

* If $x = 1$, then by (2), $t (14 \cdot 10^k + y) = 108 \cdot 10^k$ for some integer $t$. As
$8 (14 \cdot 10^k + y) \ge 8 \cdot 14 \cdot 10^k > 108 \cdot 10^k > 6 \cdot 15 \cdot 10^k > 6 (14 \cdot 10^k + y)$,
the only option is $t = 7$, but the equation $7 (14 \cdot 10^k + y) = 108 \cdot 10^k$ has no integer solutions, because $7 \nmid 108 \cdot 10^k$.

* If $x = 2$, then by (2), $t (24 \cdot 10^k + y) = 198 \cdot 10^k$ for some integer $t$. As
$9 (24 \cdot 10^k + y) \ge 9 \cdot 24 \cdot 10^k > 198 \cdot 10^k > 7 \cdot 25 \cdot 10^k > 7 (24 \cdot 10^k + y)$,
the only option is $t = 8$. The equation $8 (24 \cdot 10^k + y) = 198 \cdot 10^k$ yields $y = \frac{6 \cdot 10^k}{8} = 75 \cdot 10^{k-2} < 10^k$. Altogether $n = 2475 \cdot 10^{k-2}$.

* If $x \ge 3$, then by (3) we have $x \cdot 10^{k+1} + 4 \cdot 10^k + y \le 18 \cdot 10^k + 9y$, which yields $x \cdot 10^{k+1} \le 14 \cdot 10^k + 8y$. On the other hand $x \cdot 10^{k+1} \ge 30 \cdot 10^k > 22 \cdot 10^k > 14 \cdot 10^k + 8y$. The equations contradict each other, so no such $n$ can exist.