Cyclic Numbers

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This Demonstration shows the first 10 cyclic numbers.


A cyclic number with digits is such that its digits are shifted cyclically when multiplied by an integer up to . For example, with , ; multiplying by 6 shifts the digits of by three places: . The decimal representation of the reciprocal of has a period of maximum length . So .

For , leading zeros are needed for . For example, the second cyclic number (which comes from ) is the integer .


Contributed by: Izidor Hafner (December 2016)
Open content licensed under CC BY-NC-SA



The first 10 values of that produce maximum period decimal expansions (with digits) for are the reptend primes 7, 17, 19, 23, 29, 47, 59, 61, 97 and 109 [1, pp. 171–175].


[1] D. G. Wells, The Penguin Dictionary of Curious and Interesting Numbers, New York: Penguin Books, 1991.

[2] N. J. A. Sloane. The On-Line Encyclopedia of Integer Sequences. "Full Reptend Primes: Primes with Primitive Root 10."

[3] N. J. A. Sloane. The On-Line Encyclopedia of Integer Sequences. "Numbers with Digits Such That the First Multiples Are Cyclic Permutations of the Number, Leading 0's Omitted (or Cyclic Numbers)."

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