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Cyclic number

A cyclic number is an integer in which cyclic permutations of the digits are successive multiples of the number. The most widely known is 142857:
:142857 × 1 = 142857
:142857 × 2 = 285714
:142857 × 3 = 428571
:142857 × 4 = 571428
:142857 × 5 = 714285
:142857 × 6 = 857142
== Details ==
To qualify as a cyclic number, it is required that successive multiples be cyclic permutations. Thus, the number 076923 would not be considered a cyclic number, because even though all cyclic permutations are multiples, they are not successive multiples:
:076923 × 1 = 076923
:076923 × 3 = 230769
:076923 × 4 = 307692
:076923 × 9 = 692307
:076923 × 10 = 769230
:076923 × 12 = 923076
The following trivial cases are typically excluded:
#single digits, e.g.: 5
#repeated digits, e.g.: 555
#repeated cyclic numbers, e.g.: 142857142857
If leading zeros are not permitted on numerals, then 142857 is the only cyclic number in decimal, due to the necessary structure given in the next section. Allowing leading zeros, the sequence of cyclic numbers begins:
:(106-1) / 7 = 142857 (6 digits)
:(1016-1) / 17 = 0588235294117647 (16 digits)
:(1018-1) / 19 = 052631578947368421 (18 digits)
:(1022-1) / 23 = 0434782608695652173913 (22 digits)
:(1028-1) / 29 = 0344827586206896551724137931 (28 digits)
:(1046-1) / 47 = 0212765957446808510638297872340425531914893617 (46 digits)
:(1058-1) / 59 = 0169491525423728813559322033898305084745762711864406779661 (58 digits)
:(1060-1) / 61 = 016393442622950819672131147540983606557377049180327868852459 (60 digits)

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