142857
The number made up of the six digits that repeat in the decimal expansion of 1/7 (0.142857142857…). Multiplying it by 1 through 6 gives, in every case, the same six digits in the same order, rotated around, and it is the best-known cyclic number in base 10. Multiplying it by 7 gives 999999.
The cyclic property
Each of the products from 1 to 6 is the digit sequence of 142857 rotated to a different starting point.
- 142857 × 1 = 142857
- 142857 × 2 = 285714
- 142857 × 3 = 428571
- 142857 × 4 = 571428
- 142857 × 5 = 714285
- 142857 × 6 = 857142
This corresponds to the fact that the repeating parts of the decimal expansions of 2/7, 3/7 … 6/7 are each a rotation of 142857.
Multiplying by 7 and runs of 9s
- 142857 × 7 = 999999
- 142 + 857 = 999 (the sum of the first three digits and the last three)
- 14 + 28 + 57 = 99 (the sum when split into pairs of digits)
The property that splitting the repeating part in half and adding the halves gives a run of 9s is generalized as Midy's theorem.
Kaprekar number
142857 is also a Kaprekar number. Squaring it gives 20408122449, and splitting this into 20408 and 122449 and adding them returns the original 142857.
To verify
- Its relation to other cyclic numbers, such as those obtained from 1/17 and 1/19
References
- Mentioned in
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