> 
> Prove that there exists a natural number $n$ that has more than $2017$ divisors $d$ satisfying
> $$\tag 1 \sqrt n \le d < 1.01\sqrt n$$
> 
> 
> 

My reasoning was that $\tag 2 1.01\sqrt n-\sqrt n\ge2019$ must occur. Otherwise, there won't be enough space to fit in enough numbers in $(1)$. $(2)$ gives us that $n\geq 201900^2$. We also have that $201900^2<201900!$ so we can pick any factorial greater than or equal to $201900!$ and that satisfies the desired condition. Is this solution correct?