4. (35 points) Magician David Copperfield secretly told the same natural number to schoolchildren Misha and Masha. Masha multiplied this number by 3 several times, while Misha added 2500 to this number the same number of times. After the children compared the results of their calculations, it turned out that they had the same numbers again. Find three different numbers that Copperfield could have told Misha and Masha.
Answer: $1250, 625, 125$.

Solution. Let $x$ be Copperfield's number, and $k$ be the number of times Masha multiplied the number by 3. Then the equation $x + 2500k = x \cdot 3^{k}$ holds. From this, we get $x = \frac{2500k}{3^{k} - 1}$. By substitution, we find that $k=1, 2, 4$ work, corresponding to the values $x_{1}=1250$, $x_{2}=625, x_{3}=125$.