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Greetings again ftom North Wales!

In this post, I would like to look at a most remarkable number – 142857.

In a prior post of mine covering the subject of divisibility testing, I analysed the factors of this number, determining that it was divisible by the following factors only:-

3 x 3 x 3 x 11 x 481

Now I will look into 142857 more closely. And it is indeed a most remarkable number, as I first discovered when looking it up in Shakuntala Devi’s book Figuring: The Joy of Numbers, way back in the mists of time.

For one thing, 142857 is a cyclic number. Take a look at that happens when you multiply 142857 by the numbers 1 through 6:-

142857 x 1 = 142857

142857 x 2 = 285714

142857 x 3 = 428571

142857 x 4 = 571428

142857 x 5 = 714285

142857 x 6 = 857142

As you can see, the numbers simply rotate, each time the next larger number moving into the left most column and the remaining digits staying in order behind it.

Of course, the fun does not end there. This is what happens next:-

142857 x 7 = 999999

Now let’s look at the digits themselves, dividing them up into two groups: 1, 4, 2 and 8, 5, 7.

What is intriguing here is that a connection exists between the digits of the first half and those of the second half:-

4 x 2 = 8

1 + 4 = 5

1 + 2 + 4 = 7

Another property of 142857 draws upon the fact that 27 (33) is one of its factors. If you draw up a table containing the digits of the multiples of 142857 from 1 to 6, above, you will note that the rows and columns all add up to 27:-

1 4 2 8 5 7 = 27
2 8 5 7 1 4 = 27
4 2 8 5 7 1 = 27
5 7 1 4 2 8 = 27
7 1 4 2 8 5 = 27
8 5 7 1 4 2 = 27
== == == == == ==
27 27 27 27 27 27

Lastly, 142857 appears in the recurring decimals of 7, where once again it demonstrates its cyclic nature:-

1/7 = 0. 142857 142857 142857 …

2/7 = 0. 285714 285714 285714 …

3/7 = 0. 428571 428571 428571 …

4/7 = 0. 571428 571428 571428 …

5/7 = 0. 714285 714285 714285 …

6/7 = 0. 857142 857142 857142 …

Once again we see a relationship between 142857 and 7, a connection made all the more remarkable by the fact that 7 is not itself a factor of 142857 – as I have pointed out previously, mathematics is not immune to a little display of irony every now and again.

In all, 142857 is indeed a most remarkable number and, like 23 and 1,024 it has become one of my favourite numbers.

6 COMMENTS

  1. More fun:

    142+857 = 999
    285+714 = 999
    428+571 = 999
    571+428 = 999
    714+285 = 999
    857+142 = 999

    14+28+57 = 99
    28+57+14 = 99
    42+85+71 = 198 = 99 + 99
    57+14+28 = 99
    71+42+85 = 198 = 99 + 99
    85+71+42 = 198 = 99 + 99

    and:
    1+4+2+8+5+7 = 27; 2+7 = 9

    …more:
    142857×142857 = 20 408 122 449;
    20408+122449 = 142857

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