For each positive integer $k$ we denote by $S(k)$ the sum of its digits, for example $S(132)=6$ and $S(1000)=1$. A positive integer $n$ is said to be $\textbf{fascinating}$ if it holds that $n = \frac{k}{S(k)}$ for some positive integer $k$. For example, the number $11$ is $\textbf{fascinating}$ since $11 = \frac{198}{S(198)} ($since $\frac{198}{S(198)}=\frac{198}{1+9+8}=\frac{198}{18} = 11)$. Prove that there exists a positive integer less than $2021$ and that it is not $\textbf{fascinating}$.