2. A student thought of a number, appended 2 to it from the right, and added 14 to the resulting number. To the number thus obtained, he appended 3 from the right and added 52 to the resulting number. When the number thus obtained was divided by 60, the quotient was 6 more than the thought number, and the remainder of the division was a two-digit number written with the same digits, with the tens digit equal to the thought number. Determine the thought number.
Solution. Let $x$ be the thought number. The student first obtained the number $10 x+2$, then the number $10 x+16$, and then the number $10(10 x+16)+3$ and finally obtained the number $10(10 x+16)+3+52=10(10 x+16)+55$. When the last number was divided by 60, the quotient was $x+6$ and the remainder was $10 x+x=11 x$. According to this,

$$
10(10 x+16)+55=60(x+6)+11 x
$$

from which we get $x=5$.