Rotated Multiples

When you multiply 102564 by 4, its rightmost digit moves to the left and the other digits move one space to the right (4 102564 = 410256). Find the smallest number that has this same property when multiplied by N (where N is each integer from 2 to 9.)  Some answers are surprisingly long.

Source: Based on Macalaster College Problem of the Week #1013.


Solution
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