7.2. Does there exist a number of the form $1000 \cdots 001$ that can be divided by a number of the form $111 \cdots 11$?
7.2. Only for 11

In fact, $1000 \cdots 001=99 \cdots 9+2$, and the remainder of $99 \cdots 9$ divided by $111 \cdots 11$ is a number of the form $99 \cdots 9$ with fewer digits than the divisor. Therefore, if $99 \cdots 9+2$ can be divided by $111 \cdots 11$, then $99 \cdots 9+2$ must equal $111 \cdots 11$. This is only possible for 11.