130. Find the smallest value of the expression $\left|36^{m}-5^{n}\right|$, where $m$ and $n$ are natural numbers.
Solution. The difference $36^{m}-5^{n}$ ends with the digit 1, and the difference $5^{n}-36^{m}$ ends with the digit 9. We will prove that $\left|36^{m}-5^{n}\right|$ cannot be equal to either 1 or 9.

From the equation $36^{m}-5^{n}=1$, it follows that $\left(6^{m}-1\right)\left(6^{m}+1\right)=5^{n}$, and then the power of the number 5 is divisible by the number $6^{m}+1$, which ends with the digit 7, which is impossible. From the equation $5^{n}-36^{m}=9$, it follows that $5^{n}$ is divisible by 9, which is incorrect.

Since $36-5^{2}=11$, then 11 is the smallest value of the given expression.

Answer: 11.