13.205. If a two-digit number is divided by the sum of its digits, the quotient is 3 and the remainder is 7. If then the sum of the squares of the digits of this number is taken and the product of the same digits is subtracted from it, the original number is obtained. Find this number.
Solution.

Let $10 x+y$ be the desired number. According to the condition $\left\{\begin{array}{l}10 x+y=3(x+y)+7, \\ x^{2}+y^{2}-x y=10 x+y,\end{array}\right.$ from which $x=3, y=7$. The desired number is 37.

Answer: 37.