A truly wonderful number that I came across while going through a book
If you multiply 142857 with anything between 1 - 6, you will get the same digits circularly shifted in their positions for equal number of places, irrespective of their direction of shift
Illustrating,
142857 x 2 = 285714
142857 x 3 = 428571
142857 x 4 = 571428
142857 x 5 = 714285
142857 x 6 = 857142
So here the question is why this happens??
The book provided hint in the form that this arrangement might be understood from identifying that this is the recurring decimal digits of 1/7, which might be used in summing the digits appearing in their respective infinite series, followed by manipulating the sides of the identities thus created
But a slight research with a wiki hint helped me to understand it in an even more independent style
Critically looking at 142857 reveals that this lovely number is consisting of double digits forms that either is a multiple of 14 or > a multiple of 14 by 1
Illustrating,
considering f = 14
42 = 3f
28 = 2f
85 = 6f + 1
57 = 4f + 1
71 = 5f + 1
Accordingly, this particular arrangement may be represented in the following form
142857 = 10⁴f + 2.10²f + (4f + 1)
Interestingly,
7f + 2 = 100
These give,
2 x 142857 = 2.10⁴f + 4.10²f + (8f + 2)
= 2.10⁴f + 4.10²f + 10² + f [remembering (7f + 2) = 100]
= 2.10⁴f + (4f + 1)10² + f
= 285714
Similarly,
3 x 142857 = 3.10⁴f + 6.10²f + (12f + 3)
= 3.10⁴f + 6.10²f + 10² + (5f + 1)
= 3.10⁴f + (6f + 1)10² + (5f + 1)
= 428571
Remaining products are left solely for the interested
