7. Find the smallest natural number ending in the digit 2 that doubles when this digit is moved to the beginning.
Answer: 105263157894736842

Solution: Let's write the number in the form ***...** 2 and gradually restore the "asterisks" by multiplying by 2:

```
\(* * \ldots * * 2 \times 2=* * * . . . * * 4\)
\(* * * . . * 42 \times 2=* * * \ldots * 84\)
\(* * * . . .842 \times 2=* * * \ldots * 684\)
***... \(* 6842 \times 2=* * * \ldots * 3684\)
\(* * * . . * 36842 \times 2=* * * \ldots * 73684\)
\(* * * . . * 736842 \times 2=* * * . . . * 473684\)
\(* * * . . * 4736842 \times 2=* * * \ldots * 9473684\)
\(\cdots\)
\(105263157894736842 \times 2=210526315789473684\)
```