Derviche numbers

Multiply 142857 by 3, you get 428571, the same digits, but with a circular shift : the 1st digit has become the last.
We can rotate in the reverse direction : 142857 × 5 = 714285, the last digit has become the first.
Such a number (142857) is a "Derviche number", rotating by multiplication.
Even more, when multiplying by 2,3,4,5,6, we get all its circular shifts.

Excluding such excess, just find the smallest number, which multiplied by 2 rotates one digit right
Solution

Rotate one digit left ?
Solution

Same questions when multiplying by 3, 4, ... 9
Solution

Details

 

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