13.237. A positive integer was thought of. To its digital representation, a digit was appended on the right. From the resulting new number, the square of the thought number was subtracted. The difference turned out to be 8 times the thought number. What number was thought of and which digit was appended?
## Solution.

If the number $x$ was thought of, then by appending the digit $y$ to the right, we get the number $10 x + y$.

According to the condition $10 x + y - x^2 = 8 x \Leftrightarrow x^2 - 2 x - y = 0, \quad x = 1 \pm \sqrt{1 + y}$. Then the possible values of $y$ are: $0, 3, 8$. From this, the thought-of number is $2, 3, 4$ respectively.

Answer: appended 0, or 3, or 8; thought of respectively $2, 3, 4$.