5. Find the smallest prime number $p$ such that $n^{2}+n+11$ is divisible by $p$ for some integer $n$.

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6.II.5. Since $n(n+1)$ is an even number, the number $n^{2}+n+11$ is odd, and therefore this number is not divisible by 2. Since $n^{2}+n+11=$ $(n-1)^{2}+1+3n+9$, the number $n^{2}+n+11$ is not divisible by 3, because no perfect square can have a remainder of 2 when divided by 3. Since $n^{2}+n+11=(n-2)^{2}+2+5n+5$, by similar reasoning, this number is not divisible by 5. Further, since $n^{2}+n+11=$ $(n-3)^{2}+2+7n$, this number is not divisible by 7 because no perfect square can have a remainder of 5 when divided by 7. Finally, note that when $n=0$, the number $n^{2}+n+11$ is divisible by 11. Therefore, 11 is the smallest prime divisor among all divisors of numbers of the form $n^{2}+n+11$.