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The number 142857, which is involved in the decimal representation of 1/7, has some interesting properties, enough such it is sometimes called a magic number. Consider its multiples.
| The Multiples of 142857 | ||
|---|---|---|
| Multiplier | Multiple | Comment |
| 1 | 142857 | |
| 2 | 285714 | First two digits moved to end |
| 3 | 428571 | First digit moved to the end |
| 4 | 571428 | First four digits moved to the end |
| 5 | 714285 | First five digits moved to the end |
| 6 | 857142 | First three digits moved to the end |
| 7 | 999999 | |
| 8 | 1142856 | One deducted from the last digit and moved to the front |
| 9 | 1285713 | 13 deducted from the initial 14 and moved to the end |
Another magic number is 076923, which comes from the decimal representation of 1/13.
| The Multiples of 076923 | ||
|---|---|---|
| Multiplier | Multiple | Comment |
| 1 | 076923 | |
| 2 | 153846 | |
| 3 | 230769 | First four digits moved to the end |
| 4 | 307692 | Last digit moved to the front |
| 5 | 384615 | |
| 6 | 461538 | |
| 7 | 538461 | |
| 8 | 615384 | |
| 9 | 692307 | First two digits moved to the end |
| 10 | 769230 | First digit 0 moved to the end |
| 11 | 846153 | |
| 12 | 923076 | First three digits moved to the end |
| 13 | 999999 | |
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