8.2. Does the equation $x^{9}=2013 y^{10}$ have a solution in natural numbers $x, y$?
Answer: Yes. Solution. We will look for a solution in the form of powers of the number 2013, i.e., $x=2013^{m}, y=2013^{n}$. Then $2013^{9 m}=2013^{1+10 n}$, i.e., $9 m-1=10 n$. We can take $m=9$ (the smallest natural $m$ for which the last digit of the number $9 m$ is 1), then $n=8$.