## Task 22/65

We are looking for a natural number with the following property: The last digit is $z$; if you remove this digit and place it as the first digit before the remaining digits, the result is seven times the original number.
Since the "new number" is supposed to be seven times the original number and is derived from it by removing and prefixing the last digit, the second-to-last digit must result from multiplying the last digit by 7.

Since $7 \cdot z = 49$, the second-to-last digit is 9.

The third-to-last digit is derived by multiplying the second-to-last digit by 7, while also considering the carry-over of 4 tens from 49; it thus results from $7 \cdot 9 + 4 = 67$ to 7. The fourth-to-last digit is then derived from $7 \cdot 7 + 6 = 55$ to 5.

This process is continued until a sequence of 10 digits has appeared. This way, the smallest natural number with the required property is obtained. If the process were continued beyond this point, a periodicity would result; this shows that there are infinitely many such numbers.

The number found in this way is 1014492753623188405797.

If the last digit is removed and placed at the beginning, the resulting number is 7101449275362318840579, which is indeed seven times the original number.