1. Person A writes a four-digit number $A$ on the blackboard, and Person B adds a digit before and after this four-digit number to form a six-digit number $B$. If the quotient of $B$ divided by $A$ is 21 and the remainder is 0, find the values of $A$ and $B$.
1. $A=9091, B=190911$.

Let the digits added by Yi before and after $A$ be $x$ and $y$ ($1 \leqslant x \leqslant 9, 0 \leqslant y \leqslant 9$). Then
$$
B=10^{5} x+10 A+y \text {. }
$$

According to the problem, $B=21 A$, which means
$$
\begin{array}{l}
10^{5} x+10 A+y=21 A \\
\Rightarrow 10^{5} x+y=11 A .
\end{array}
$$

Notice that, $11 A \equiv 0(\bmod 11)$.
Thus, $10^{5} x+y \equiv-x+y \equiv 0(\bmod 11)$.
Therefore, among the possible values of $x$ and $y$, only $x=y$.
By testing $x$ and $y$ simultaneously equal to $1 \sim 9$, we find that when $x=y=1$, $A$ is a four-digit number 9091.