41. After persistent and prolonged searches, Andrey found a two-digit number that does not end in 0 and has the property that, by inserting a zero between the tens and units digits, a three-digit number is obtained that is divisible without a remainder by the original two-digit number. When he showed his result to the teacher, the teacher, praising the boy, noted that not only the number Andrey found has this property. Find all such two-digit numbers. How many are there?

![](https://cdn.mathpix.com/cropped/2024_05_21_00604dc020e3721fc1f4g-046.jpg?height=414&width=434&top_left_y=987&top_left_x=340)

## Search for solutions
41. Let the desired number be $\overline{a b}$, then the three-digit number is $\overline{a o b}$. According to the condition, $\overline{a b} \cdot n = \overline{a o b}$ 146
or $(10 a + b) \cdot n = 100 a + b$, from which $10 a(10 - n) = b(n - 1)$. Therefore, $110 - n$, from which $n > 5$. Since the left side of the equation is divisible by two, then $(n - 1) \vdots 2$, so $n$ is odd. It remains that $n = 7$ or $n = 9$. For $n = 7$ we have $2 a \cdot 3 = 6, a = 1$. We get $15 \cdot 7 = 105$, which satisfies the condition. For $n = 9$ we have $2 a = 8$, $a = 4$. We get $45 \cdot 9 = 405$, which satisfies the condition. Thus, there are 3 numbers: 15, 18, and 45, which have the specified property.

![](https://cdn.mathpix.com/cropped/2024_05_21_00604dc020e3721fc1f4g-148.jpg?height=418&width=409&top_left_y=656&top_left_x=375)

Greatest Common Divisor and Least Common Multiple