## 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?

### 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).