For example, a number in the form of $42 \cdots$ multiplied by 2, with 42 moved to the end, find this number.
Given the theorem $x=42, k=2, c=2, \omega=\frac{x}{10^{2}-2}$ $=\frac{42}{98}=0.42857^{\circ} \mathrm{i}$, therefore, the numbers that meet the condition are $428571,428571428571, \cdots$