13.421 If a two-digit number is divided by a certain integer, the quotient is 3 and the remainder is 8. If the digits in the dividend are swapped while the divisor remains the same, the quotient becomes 2 and the remainder is 5. Find the original value of the dividend.
Solution. Let $\overline{x y}$ be the desired number, $z$ be the divisor. Then, according to the condition,
\[
\left\{\begin{array}{l}
10 x+y=3 z+8 \\ 
10 y+x=2 z+5
\end{array}\right.
\]

By eliminating $z$ from these equations, we have: $17 x-28 y=1 \Leftrightarrow 28(x-y)=1+11 x$.

But $12 \leq 1+11 x \leq 100$. In this range, only the numbers $28, 56, 84$ are divisible by 28. We directly verify that only $56=1+11 x \Rightarrow x=5 \Rightarrow y=3$.

Answer: 53.