Find the smallest natural number k such that between k different and in pairs relatively prime natural numbers less than 2021 there's a prime number

Find the smallest natural number $k$ such that between $k$ different and in pairs relatively prime natural numbers less than $2021$ there's a prime number.

Honestly have no idea how to start here so if anyone could share a hint that'd be great.

Hint: Suppose that you have a maximal set $S$ (in terms of cardinality) of non-prime natural numbers such that any two numbers in $S$ are relatively prime. Then, show that you can obtain a set $S'$ of non-prime natural numbers satisfying the given condition such that $S'\subseteq T:=\{1\}\cup\{p^2<2021|p \text{ is prime}\}$ and $|S|=|S'|$. To see this, take an element $s\in S$ that is not in $T$. Then, $s={p_1}^{r_1}{p_2}^{r_2}\ldots{p_n}^{r_n}$ for some $n\geq1$, $\;\;r_1,\ldots,r_n \in\mathbb{N}$ and primes $p_1<p_2<\ldots<p_n$. Now, observe that you could replace $s$ by $p^2$ and have all the required properties satisfied. Inductively, you can obtain $S'$. Now observe that $T$ is a set of non-prime natural numbers such that any two numbers in $T$ are relatively prime and $|S|=|S'|\leq|T|$. Infer that $k=|T|+1$.