Writing It Up · Superchallenge problem

The proof of $24 \mid p^2 - 1$ combines "odd $\Rightarrow 8 \mid n^2 - 1$" and "$3 \nmid n \Rightarrow 3 \mid n^2 - 1$". A student adds "$5 \nmid n \Rightarrow 5 \mid n^2 - 1$" to claim $120 \mid n^2 - 1$ whenever $\gcd(n, 30) = 1$. What is the smallest $n > 1$ with $\gcd(n, 30) = 1$ refuting it?