142857
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The number 142857 has the curious property that, if it is multiplied by any number from 1 to 6, the result is a number with the same digits in the same order (a cyclic permutation).
For example:
- 142857 × 1 = 142857
- 142857 × 2 = 285714
- 142857 × 3 = 428571
- 142857 × 4 = 571428
- 142857 × 5 = 714285
- 142857 × 6 = 857142
This property is closely related to the facts that
- 1/7 = 142857/999999
and that 1/7 expressed as a decimal is
- 0.142857142857142857142857142857...
Investigating the number 142857 can lead a student into an understanding of repeating decimals and rational numbers.
Similar Numbers
142857 is the only multi-digit decimal number with this property. However, if leading zeros are allowed on a number, some other examples are:
- 0588235294117647 (16 digits)
- 052631578947368421 (18 digits)
- 0434782608695652173913 (22 digits)
- 0344827586206896551724137931 (28 digits)
- 0212765957446808510638297872340425531914893617 (46 digits)
In octal (base-8 notation), the numbers 25 and 1463 have this property. There are no examples in hexadecimal (base-16).