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1. CAROUSEL NUMBERS (the appetizer problem)
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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
digits.
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
---------------- ----------------
1176470588235294
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...)