Note: This is an old post on an old blog. View the current version of this page on: http://neothebrute.blogspot.com/2008/04/mystery-of-number-7.html

One fine day I had simply no work in life, so I decided to calculate PI (22/7 = 3.142857…) to the millionth decimal place. Needless to say, I didn’t reach that mark, but I was astounded to see that not only did PI follow the sequence of 142857 after the decimal place; EVERY number possible did the same (except obviously multiples of 7, which don’t need the decimal point). Every number that I divided with 7 followed the circular chain of 142857 after the decimal place, some starting with 4 immediately after the decimal place and going through the following 2857 before restarting the 142857 sequence etc. But essentially everyone had the same chain, starting at different points in the circular chain. Observe:

22 / 7 = 3.{142857,142857,142857,142857,142857…}
227 / 7 = 32.{42857}{142857,142857,142857,142857…}
890 / 7 = 127.{142857,142857,142857,142857,142857…}
22792279 / 7 = 3256039.{857}{142857,142857,142857,14287…}

So I set myself an objective: Find out why numbers when divided by 7 present the same pattern of 142857. There is no conspiracy theory or ‘number of God 7′ in this. Maybe people have already figured out the solution to this problem, which proves that I’m not the only one with no work and loads of time to kill.

My explanation:
When any number is divided by 7, at each step of the long division method the remainder will be from the following sequence, maybe in a circularly, rotated way: 1, 3, 2, 6, 4, 5. Observe:

(By Long Division method)
22 / 7 = 3, and remainder = 1
put decimal point and bring down a ‘zero’ => 1 is now 10
10 / 7 = 1, and remainder = 3
bring down another ‘zero’ => 3 is now 30
30 / 7 = 4, and remainder = 2
bring down another ‘zero’ => 2 is now 20
20 / 7 = 2, and remainder = 6
bring down another ‘zero’ => 6 is now 60
60 / 7 = 8, and remainder = 4
bring down another ‘zero’ => 4 is now 40
40 / 7 = 5, and remainder = 5
bring down another ‘zero’ => 5 is now 50
50 / 7 = 7, and remainder = 1
bring down another ‘zero’ => 1 is now 10
… and the cycle begins again. Remainders thus cycle in the sequence 1, 3, 4, 6, 4, 5.

Also observe that this remainder sequence includes all the numbers less than 7, which means that whatever be the dividend, the remainder at each step of division HAS to be one of these (as 7 or greater can’t be a remainder while dividing with 7!), and therefore the answer will start at some point inside this quotient chain. Since these are all the possible numbers less than 7, this means that any number when divided by 7 WILL produce the remainders chain of 1,3,2,6,4,5 and thus produce the quotient chain of 142857. Hence this mystery has been explained.