10. $(8,9)$ Find the smallest possible value of $\left|2015 m^{5}-2014 n^{4}\right|$, where $m$ and $n$ are natural numbers.
Answer: 0.

Solution: Consider numbers of the form $m=2014^{a} \cdot 2015^{b}$ and $n=2014^{c} \cdot 2015^{d}$. Then $\left|2015 m^{5}-2014 n^{4}\right|=\left|2014^{5 a} \cdot 2015^{5 b+1}-2014^{4 c+1} \cdot 2015^{4 d}\right|$. This value equals 0 in the case $5 a=4 c+1,5 b+1=4 d$. It is not difficult to find such numbers, for example, $a=c=1, b=3, d=4$.