10. For some positive integer $n$, the fraction $\frac{n+2}{3 n^{2}+7}$ is not in its simplest form, then the minimum value of $n$ is $\qquad$ .
Let $n+2=a d, 3 n^{2}+7=b d,(a, b)=1$, then $3(n+2)^{2}-\left(3 n^{2}+7\right)=\left(3 a^{2} d-b\right) d=12 n+5$, at this point $\left(3 a^{2} d-b, a\right)=1 \Rightarrow(12 n+5, n+2)=d$.
Thus, from $(12 n+5, n+2)=(12(n+2)-(12 n+5), n+2)=(19, n+2)=19=d \Rightarrow n_{\min }=17$.