13.239. To the digital record of a certain two-digit number, the same number was appended to the right, and from the resulting number, the square of the intended number was subtracted. The difference was divided by $4 \%$ of the square of the intended number; in the quotient, half of the intended number was obtained, and in the remainder - the intended number. What number is intended?

To the digital record of a certain two-digit number, the same number was appended to the right, and from the resulting number, the square of the intended number was subtracted. The difference was divided by $4 \%$ of the square of the intended number; in the quotient, half of the intended number was obtained, and in the remainder - the intended number. What number is intended?
Solution.

Let $a$ be the number thought of. Then the new number is $100a + a = 101a$.

According to the condition, $101a - a^2 = 0.04a^2 \cdot \frac{1}{2}a + a$, from which $a = 50$.

Answer: 50.

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-1167.jpg?height=303&width=293&top_left_y=81&top_left_x=94)

Fig. 13.3