13.305. Find two two-digit numbers $A$ and $B$ under the following conditions. If the number $A$ is written before the number $B$, and the resulting four-digit number is divided by the number $B$, the quotient is 121. If the number $B$ is written before the number $A$, and the resulting four-digit number is divided by $A$, the quotient is 84 and the remainder is 14.
Solution.

According to the condition, we have the system $\left\{\begin{array}{l}100 A+B=121 B, \\ 100 B+A=84 A+14,\end{array}\right.$ from which $A=42, B=35$.

Answer: 42 and 35.