13.449 The desired three-digit number starts with the digit 1. If it is erased and then written as the last digit of the number, the new three-digit number obtained will be greater than the desired number by $9 a^{1 / \lg a}$. Find this number.
Solution. Let $\overline{1 x y}$ be the desired number. According to the condition

$$
\begin{aligned}
& \overline{1 x y}=\overline{x y 1}-9 \cdot a^{\log _{a} 10}=\overline{x y 1}-90 \text { or } 100+10 x+y=100 x+10 y+1-90 \Leftrightarrow \\
& \Leftrightarrow 21=10 x+y \Rightarrow x=2, y=1
\end{aligned}
$$

Answer: 121.