34 Find a positive integer $n$, such that each number $n, n+1, \cdots, n+20$ is not coprime with 30030.
34. Since $30030=2 \times 3 \times 5 \times 7 \times 11 \times 13$, if we take $N=210 k$, then $N$ and $N \pm r$ are not coprime with 30030, where $r$ is a number in $2,3, \cdots, 10$. Now consider the numbers $N \pm 1$. We choose $k$ such that
$$210 k \equiv 1(\bmod 11) \text { and } 210 k \equiv-1(\bmod 13),$$

The first condition requires $k \equiv 1(\bmod 11)$, set $k=11 m+1$,
The second condition requires
$$210(11 m+1) \equiv-1(\bmod 13)$$

Solving this, we get
$$m \equiv 4(\bmod 13)$$

Therefore, let $k=45$, then the 21 numbers $9440,9441, \cdots, 9460$ are not coprime with 30030, so taking $n=9440$ is sufficient.