Let n be a positive integer and consider 1/n, as an infinite decimal.
Under some circumstances, when one multiplies the repeating pattern
of digits in such a decimal representation, by the numbers 2,...,
n-1, the digits "cycle" around. (Note: multiplying this
collection of digits by n yields all 9"s.)
Example:
1/7 = .14285712857...
Now:
2 x 142857 = 285714
3 x 142857 = 428571
4 x 142857 = 571428
5 x 142857 = 714285
6 x 142857 = 857142
Sometimes one has to use an initial zero as part of the pattern
of digits that "cycle" around. Gary Klatt (University
of Wisconsin - Whitewater) has used the attractive term "carousel"
numbers for those n for which this phenomenon occurs. For some
numbers one gets not a full "cycle" but some "partial"
cycling.
Problem:
1. Which numbers are carousel numbers?
2. Which numbers show some partial cycling?
3. What happens for numbers expressed in bases other than 10?
4. How is the length of the cycle related to the nature of the
number n?