3. Find the smallest value of the expression $\left|36^{m}-5^{n}\right|$, where $m$ and $n$ are natural numbers.
Solution. We consider two cases:
a) $36^{m}>5^{n}$. The last digit of the number $36^{m}-5^{n}$ is 1. Let $36^{m}-5^{n}=1$. Then $5^{n}=\left(6^{m}+1\right)\left(6^{m}-1\right)$, which is not possible, since 5 is not a divisor of $6^{m}+1$.

Let $36^{m}-5^{n}=11$. Obviously, for $m=1, n=2$ the equality is satisfied.
b) $36^{m}<5^{n}$. The last digit of the number $5^{n}-36^{m}$ is 9. Let $5^{n}-36^{m}=9$. This equation has no solution in the set of natural numbers since 9 is not a divisor of $5^{n}$.

Therefore, the smallest value of $\left|36^{m}-5^{n}\right|$ is 11.