Problem 2. Determine the integers $m$ and $n$ for which

$$
\frac{147 \cdot 7^{3}+28 \cdot 7^{4}}{7^{n}}=7^{m} \text { and } \frac{5^{4} \cdot 25^{2}}{125^{m} \cdot 5^{-10}}=5^{n}
$$
Solution. With elementary transformations, it is easily obtained that the given equations are equivalent to the equations

$$
7^{6-n}=7^{m} \text { and } 5^{18-3 m}=5^{n}
$$

from which we get $6-n=m$ and $18-3 m=n$. If we substitute the second equation into the first, we get

$$
6-(18-3 m)=m, \text { i.e. } m=6
$$

so, $n=18-3 \cdot 6=0$