jmc

Since 2, 3, and 5 divide $30N$ but not $7$, they do not divide $30N+7$. Similarly, 7 only divides $30N+7$ if 7 divides $30N$, which means $N$ must be a multiple of 7 for 7 to divide it. Since no number less than 11 divides $30N+7$ while $N<7$, we only need to check when $30N+7\ge 11^2$. When $N=4$, $30N+7=127$ is prime. When $N=5$, $30N+7=157$ is prime. However, when $N=\boxed{6}$, $30N+7=187=11\cdot 17$ is composite.