13.173. If an unknown two-digit number is divided by the number represented by the same digits but in reverse order, the quotient is 4 and the remainder is 3. If the sought number is divided by the sum of its digits, the quotient is 8 and the remainder is 7. Find this number.
Solution.

Let $10 x+y$ be the desired number. Then $\left\{\begin{array}{l}10 x+y=(10 y+x) \cdot 4+3, \\ 10 x+y=(y+x) \cdot 8+7,\end{array}\right.$ from which $x=7, y=1$. The desired number is 71.

Answer: 71.