Show that at least one of these $15$ numbers is prime.

I think your solution is basically correct, although your expression is sometimes a bit hard to understand (perhaps due to English being a foreign language to you?).

For a shorter (but essentially equivalent) solution, let $x_1, \dots, x_{15}$ be $15$ integers in the interval $(1, 1998)$ such that none of them is prime.

For each $i$, let $p_i$ be the smallest prime factor of $x_i$. We have $p_i < \sqrt {1998} < 47$, thus each $p_i$ is among the first $14$ prime numbers.

Therefore pigeon-hole principal tells us that there exist $i \neq j$ such that $p_i = p_j$, which implies that $x_i$ and $x_j$ are not coprime.