124. To the number 43, add one digit to the left and one digit to the right so that the resulting four-digit number is divisible by 45. Find all solutions.
$\triangle$ Let's denote the unknown digits by $a$ and $b$. Then the four-digit number can be written as $\overline{a 43 b}$.

According to the divisibility rule for 5, $b=0$ or $b=5$. Let's consider both cases.

1) Suppose $b=0$.

The number $\overline{a 430}$ is divisible by 9 if and only if the sum of its digits, which is $a+7$, is divisible by 9. Hence, $a=2$.

2) Suppose $b=5$.

Similarly, we find that $a=6$.

Answer: The four-digit number is 2430 or 6435.