A block that turns round: 142857
Divide 1 by 7. The block 142857 repeats. Multiply that block by 3. What do you get?
- Dividing 1 by 7 leaves remainders 1, 3, 2, 6, 4, 5 in turn. They go round in a ring.
- Each arrow carries the digit written on that step.
- Start at remainder 1. The digits read 142857.
- Start at remainder 3. The digits read 428571.
The rule
The answer is 142857 × 3 = 428571. It has the same six digits, starting at a new place.
Take 142857 times each of 1 to 6. Every product has the same six digits. They go round in the same order, starting at a different digit. A block like this is called cyclic.
Rule: Times 1 to 6, 142857 gives the same six digits, shifted.
| Fraction | Digits after the point | Same as 142857 times |
|---|---|---|
| 1/7 | 142857 | 1 |
| 3/7 | 428571 | 3 |
| 2/7 | 285714 | 2 |
| 6/7 | 857142 | 6 |
| 4/7 | 571428 | 4 |
| 5/7 | 714285 | 5 |
Why it happens
Dividing by 7 leaves six remainders: 1, 3, 2, 6, 4, 5. Then 1 comes back, so they go round in a ring. All six possible remainders are in this one ring.
Dividing 3 by 7 starts the ring at remainder 3. The division then goes round the same ring. So its digits are the same six, started at a new place.
Not every bottom number does this. 1 ÷ 13 has the block 076923. 2 ÷ 13 has a different block, 153846. A division by 13 can leave twelve remainders, 1 to 12. But they form two rings of six.
| Bottom number | Block length | One ring? |
|---|---|---|
| 7 | 6 | yes |
| 17 | 16 | yes |
| 19 | 18 | yes |
| 13 | 6 | no, two rings |
Watch out!
Times 7 gives 999999, not a shift. The pattern holds for 1 to 6.
Slide the multiplier. Watch the six digits.
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