The Magical Number 142857

So, what's the big deal about a number, you would say! Well, not if this number graces the cover of a book called "The Joy of Numbers" by Shakuntala Devi.

The great thing about this number is when multiplied by an integer you have an 86% chance of the digits just kind of shifting around. Look at this:

142857 = 142857 x 1
285714 = 142857 x 2
428571 = 142857 x 3
571428 = 142857 x 4

See the pattern? All the numbers on the left are just the digits 142857, just in different orders, BUT, 142857 x 7 = 999999.... An implication of the fact that 1/7 = 0.142857 repeating. There are tricks to make the trick continue working for much larger multiples... but you can figure those out on your own... does this have application? Definitely not, but it will help you remember the decimal equivalents of fraction multiples of 7.

1/7 = 14.2857%
2/7 = 28.5714%
3/7 = 42.8571%
4/7 = 57.1428% and so on...

It can be very useful in calculations involving a multiple of 7 in the denominator. BTW, remember the irrational number pi? It is approximated as 22/7, which is a rational number. A rational number must be terminating or recurring decimal. And can you now guess the exact value of 22/7? Yes, that's right...its 3.142857 recurring! Of course, pi value is non-recurring and non-terminating...I heard someone worked hard on a program to find its value to trillion places.

Now for some interesting stuff that I found related to this number...and I quote

"The tune "Further and In Between" takes its basis from the cyclical number 142,857. Cyclical numbers are actually the inverse of certain prime numbers. For example, the number 142,857 represents the first six repeating digits of 1/7 in decimal form (0. 142857142857142857...). Also, a cyclical number with n digits, when multiplied by any whole number from 1 through n, will generate a product consisting of the same digits, which in every case except for when the multiple is 1, will be in a different sequence. Mahanthappa took various forms of his chosen number (142,857; 428,571; etc.), strung them end-to-end, and assigned a musical pitch to each digit. Allowing some leeway in melodic direction—tweaking pitch as needed—and working in some funk-based, start-and-stop rhythms, he constructs a jagged, leaping melody as visceral as it is intellectual."

You may read the whole article here about Rudresh Mahanthappa who brought maths into the music room.

Concepts on remainder theorem

Fermet’s theorem

If ‘a’ and ‘p’ are two numbers such that p is prime number then

( a) ^p-1 divided by p will always leave remainder 1

For example 16^38 when divided by 7 then by ferments 16^6n divided by 7 will give remainder 1

further 16^2 divided by 7 will give remainder 4.

Euler’s theorem

When ‘a’ and ‘p’ are coprime and p is not prime number than a ^E divided by ‘p’ will leave remainder 1 , given

E = p( 1- 1/x) (1- 1/y)

Where x and y are prime factors of number, note we are not concerned with powers of x and y.

For example E for hundred will be
100 = 2^2 * 5^2

E = 100(1-1/2) ( 1- 1/5) = 40

So 3^ 80 / 100 will give remainder 1 by Euler’s theorem