I was trying to solve the following problem:
Find a number in base $10$, which when multiplied by $2$, results in a number which is a cyclic shift of the original number, such that the last digit (least significant digit) becomes the first digit.
I believe one such number is $105263157894736842$
This I was able to get by assuming the last digit and figuring out the others, till there was a repeat which gave a valid number.
I noticed that each time (with different guesses for the last digit), I got an $18$ digit number which was a cyclic shift of the one I got above.
Is there any mathematical explanation for the different answers being cyclic shifts of each other?