Katka thought of a five-digit natural number. On the first line of her notebook, she wrote the sum of the thought number and half of the thought number. On the second line, she wrote the sum of the thought number and one fifth of the thought number. On the third line, she wrote the sum of the thought number and one ninth of the thought number. Finally, she added all three numbers she had written and wrote the result on the fourth line. To her surprise, she found that the number on the fourth line was the cube of a certain natural number.

Determine the smallest number Katka could have thought of at the beginning.

(L. Růžičková)

Hint. Express the sum on the fourth line in terms of the originally thought number.
If we denote the five-digit number Katka thought of by $x$, then the numbers written on the first three lines were $x+\frac{1}{2} x=\frac{3}{2} x, x+\frac{1}{5} x=\frac{6}{5} x$ and $x+\frac{1}{9} x=\frac{10}{9} x$. The sum on the fourth line was equal to

$$
\left(\frac{3}{2}+\frac{6}{5}+\frac{10}{9}\right) x=\frac{343}{90} x
$$

This result is supposed to be the cube of some natural number, so it must be a natural number itself. Since the numbers 343 and 90 are coprime, $x$ must be a multiple of 90. Since 343 is the cube of 7, $\frac{1}{90} x$ must be the cube of some natural number.

The smallest multiple of 90 that is a five-digit number is $10080=90 \cdot 112$; therefore, $\frac{1}{90} x \geqq 112$. The smallest cube of some natural number that is greater than or equal to 112 is $125=5^{3}$; therefore, $\frac{1}{90} x=125$.

The smallest number Katka could have thought of is $90 \cdot 125=11250$.