[Decimal numeral system]

Find all natural numbers that increase by 9 times when a zero is inserted between the units digit and the tens digit.

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Let's write our number in the form $10 a+b$, where $b$ is the units digit. We get the equation $100 a+b=9(10 a+b)$. From this, $10 a=8 b$, i.e., $5 a=4 b$. Thus, $b$ is divisible by 5. Considering the two cases $b=0, b=5$, we get the unique answer: 45.