Assume X is a positive integer. If you divide 1/X, you will get a number that eventually becomes periodic: 1/9= 0.111..., 1/4= 0.25000..., and so on. Let's call numbers like 1/9 "pure" periodic, since the fractional part is formed just by the periodic part.
Prove that:
1. For all X, you will get a periodic part, and its length will be less than X.
2. If X is even, 1/X cannot be "pure". What happens if X is odd?
3. For some X, 1/X is "pure", the period length is even, and you can split the period in two halves that sum up to all nines. For example, 1/7=0.142857 142857... and 142+857=999. Which are these X values?
Am is understanding the 'pure' concept?
1/6 is not pure because the 1 before the repeating 6's? .1666...
1/33 is pure because the are no digits before the repeating 03? .030303...
Part 1 is simple enough. If you divide 1 by X you get a remainder after figuring each digit. Eventually you have to repeat one. There are only X1 digits that have not been used already (1 is used at the beinning) so once you get one of these numbers you will repeat.
Part 2 I can see but not prove yet:
The number will be pure if it does not have 2 or 5 as a factor.
It also appears that whichever number of 2's or 5's appears most in the factorization of X is the number of digits before the repeated part begins.
For example 120= 2^3*3*5 which has three 2's but only one 5.
1/120 leads with 3 unrepeated digits: .008333...

Posted by Jer
on 20061016 11:09:06 