Zuzka wrote a five-digit number. When she appended a one to the end of this number, she got a number that is three times larger than the number she would get if she wrote a one before the original number.

Which five-digit number did Zuzka write?
A five-digit number written as $\overline{a b c d e}$ represents the value $10000 a + 1000 b + 100 c + 10 d + e$, which we denote as $x$. When we write a one in front of this five-digit number, i.e., $\overline{1 a b c d e}$, we get a number that is 100000 greater, i.e., the number $x + 100000$. When we write a one after the unknown number, i.e., $\overline{a b c d e 1}$, we get the number $10 x + 1$. According to the problem, $10 x + 1 = 3 \cdot (x + 100000)$, from which $7 x = 299999$. The number on the right is indeed divisible by 7, so Zuzka wrote $x = 299999 : 7 = 42857$.

Another solution. Alternatively, the problem can be solved by multiplying "from the back" according to the following guide:

$1 a b c d e$

3

abcde 1

We calculate $3 \cdot e = (2) 1$, so $e = 7$.

| $1 a b c d 7$ |
| ---: |
| $\cdot 3$ |
| $a b c d 71$ |

We calculate $3 \cdot 7 = 21$, remember 2, then $3 \cdot d + 2 = (1) 7$, so $d = 5$. Similarly, we determine all the digits of the desired number step by step.