### Mathematics Research Projects

**Project 23 : Carousel Numbers**

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?

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### The End

**Joseph Malkevitch**

*Department of Mathematics and Computing*

York College (CUNY)

Jamaica, New York 11451-0001

email: malkevitch@york.cuny.edu

(Comments and results related to the project above are welcome.)

**Acknowledgements**

Some of this work was prepared with partial support from the National
Science Foundation (Grant Number: DUE 9555401) to the Long Island
Consortium for Interconnected Learning (administered by SUNY at
Stony Brook, Alan Tucker, Project Director).