30. Find the least prime factor of
$$
1 \underbrace{0000 \cdots 00}_{2010 \text {-many }} 1 .
$$
30. Ans: 11.
Clearly 11 is a factor and $2,3,5$ are not. We only need to rule out $7.10^{2011}+1 \equiv 4(\bmod 7)$ because $10^{3} \equiv-1(\bmod 7)$.