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Carousel Numbers:
A Lead-in to Number Theory

Gary Klatt


The number D = 076923 behaves a bit like a carousel number. It has 6
digits (again we need the leading 0), and half of the multiples
from 1D, 2D, 3D, ..., 12D are cyclic permutations of D.

For example, 3D = 230769, and 4D = 307692 give the same digits in the same
order. However 2D = 153846, and 5D = 384615 do not. (Note 5D is a cyclic
permutation of 2D. Hmmmmm... check out the other multiples)

So this number, D = 076923, is sort of "half" a carousel number. Not all
the beginning multiples produce cyclic permutations, but half do.

What is true about D = 076923 is that all of the cyclic permutations,
769230, 692307, etc are multiples of D. (Check that.)

So we might define a PARTIAL CAROUSEL NUMBER to be an integer (perhaps
with leading 0's) with the property that all of its cyclic permutations
are multiples of the number.

Again there is a connection with periodic decimals. 1/13 = .076923...
repeating with a period of 6 (half the max. potential period of 12).

Looking at the powers of 10 in U13 (which are the same as the
remainders in the long division process to find the decimal for 1/13)
we get          10 = 10 (mod 13)
                100 = 9 (mod 13) - subtract 7 13's = 91
                10^3 = 90 = 12 (mod 13)
                10^4 = 120 = 3 (mod 13)
                10^5 = 30 = 4 (mod 13)
                10^6 = 40 = 1 (mod 13).
These are exactly the multiples that "work": 10D, 9D, 12D, 3D, 4D,
(and 1D) give the 6 cyclic permutations of the digits of D.

Further, they are in the order of how much cycling happens:
        10D = 769230  cycle or "left shift" by 1
         9D = 692307    left shift by 2 from D = 076923
        12D = 923076    left shift by 3

So we are on to something here. We can find more interesting numbers
by not insisting that all the multiples 2X, 3X, ...dX be cyclic
permutations, but only that all cyclic permutations be multiples.

Alas, the inclusion of leading 0's develops into a real problem.

For example, 37 is not a partial carousel number because 73 is not
a multiple of 37. But 037 is!
        (shift left 1) 370 = 10 times 37
        (shift left 2) 703 = 19 times 37

It turns out that because 37 divides 999 (3 nines) we need the leading
zero to have a 3 digit partial carousel number.


Idea of the proof: Nearly every number divides a Nines number. The
length of the Nines number needed is related to the period of 1/n.
Add enough leading 0's to match the length of that Nines number. You
will now have a partial carousel number.

EXAMPLE   Let n = 123. Factoring, n = 3 x 41. It turns out 1/3 has
        period 1 and 1/41 has period 5, so 1/123 has period 5. We
        claim 00123 has the partial carousel property. (We have added
        enough leading 0's to bring the length to 5.)

Maybe partial carousel numbers aren't all that interesting after all.
Oh well, you can't win them all.

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