A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the six-digit number 142857, whose first six integer multiples are

To qualify as a cyclic number, it is required that consecutive 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 consecutive integer multiples:

The following trivial cases are typically excluded:

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:

Cyclic numbers are related to the recurring digital representations of unit fractions. A cyclic number of length L is the digital representation of

Conversely, if the digital period of 1/p (where p is prime) is

then the digits represent a cyclic number.

For example: