13(1254). To a certain two-digit number, a one was added to the left and right. As a result, a number was obtained that is 23 times greater than the original. Find this two-digit number.
Solution. Let the desired two-digit number be $\bar{a} \bar{b}$. According to the problem, we form the equation: $\overline{\mid a b 1}=23 \cdot \overline{a b}$. We have $1000+$ $+10 \overline{a b}+1=23 \cdot \overline{a b}$, from which $13 \cdot \overline{a b}=1001, \overline{a b}=77$.

Answer: 77.

Remark 1. The unknown two-digit number can also be found by selecting its digits, if we take into account that only by multiplying 7 by 3 we get a number ending in one.

Remark 2. It is advisable to consider problems 805-808 from the textbook [4] simultaneously with this problem.

$\triangle 14$ (1255). In a two-digit number, one digit was erased. The resulting number is 31 times smaller than the original. Which digit and in which number was erased?

Solution. From the two-digit numbers that are multiples of $31 (31, 62, 93)$, we select the numbers that satisfy the condition of the problem: after erasing one digit, we get a number that is 31 times smaller than the original. This condition is satisfied by all two-digit numbers that are multiples of 31. Indeed, by erasing the tens digit in them, we will have the numbers 1, 2, and 3, which are 31 times smaller than the numbers 31, 62, and 93, respectively.

Answer: $31, 62, 93$; the first digit needs to be erased.

$\triangle 15$ (1256). The first digit of a three-digit number is 8. If this digit is moved to the last place, the number increases by 18. Find the original number.

Solution. Method I. Let $a$ be the tens digit of the desired number, and $b$ be its units digit. Then, according to the problem, we have:

$$
\begin{gathered}
\overline{a b 8}-\overline{8 a b}=18, \text { from which } 10 \overline{a b}+8-800-\overline{a b}=18, \\
9 \overline{a b}=810, \overline{a b}=90
\end{gathered}
$$

the original number is 890.

Method II. The problem can be easily solved by selecting digits, rephrasing it as follows: find out which digits are represented by the letters, if each letter represents only one digit:

![](https://cdn.mathpix.com/cropped/2024_05_21_98932225530737b82071g-048.jpg?height=157&width=417&top_left_y=1966&top_left_x=817)

from which $B=0, A=9$.

Remark. Before solving problems 13 and 15 using equations, it is advisable to offer students the following auxiliary problems.