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Carousel Numbers:
A Lead-in to Number Theory

Gary Klatt

1.     CAROUSEL NUMBERS  (the appetizer problem)

N = 142857 has a neat property: 

       142857                        142857
           x2                            x3
       ------                        ------
       285714                        428571

We see the same digits and in the same order as the
original 142857 but just cycled around. 

Multiplying 142857 by 4, 5 and 6 gives the same result -
the other three possible cyclic permutations of the 6

Try 4łN, 5łN or 6łN on your calculator. What is 7łN?

We call the number 142857 a CAROUSEL NUMBER.

The next carousel number we know is

       M = 0588235294117647  

It has 16 digits and we need the leading zero to make it
work. Try multiplying it by 2 to see why.

       0588235294117647         0599235294117647
                     x2                      x
       ----------------         ----------------

Here all the multiples from 2 through 16 produce cyclic
permutations of the digits of M.

Note this is too large a number for most calculators to
handle. Do today's students still know how to multiply by
hand? Do you? (Prove it by trying another case above.)

Some natural questions arise:

       ARE THERE MORE?                                        (Yes, lots)

       HOW MANY?                                          (No one knows!)

       CAN ONE HAVE 2 DIGITS? ... 3 DIGITS?                 (No. Try it!)

       HOW DO WE FIND THEM?                                  (Read on...)

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