0.1 Find the smallest positive integer $\mathrm{x}$ that satisfies $11^{2021} \mid 5^{3 x}-3^{4 x}$ (40 points)
Solution:
The above equation $\Leftrightarrow 11^{2021} \mid 125^{x}-81^{x}$
By the Lifting The Exponent (LTE) lemma, $v_{11}\left(125^{x}-81^{x}\right)=v_{11}(125-81)+v_{11}(x)$
Thus, the equation $\Leftrightarrow v_{11}(x) \geq 2020$
Therefore, the minimum value of $x$ is $11^{2020}$