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?

We are told that "1/X s pure, the period length is even, and you can split the period in two halves that sum up to all nines."

If Z is the period length, then for these numbers, 10^(Z/2)(1/X)+1/X = Y, a positive integer snce the two halves sum up to all nines, and 10^(Z/2) offsets so the first numbers of each half line up. Thus, we can rewrite this as 10^(Z/2)+1=XY. It's easy to see that these will never be divisible by 2, 3 or 5.

For the other numbers, if 10^(Z/2)+1 is divisible by X, then so is (10^(Z/2)-1)(10^(Z/2)+1) which is 10^Z-1. Since 1/X is pure, it must be expressible as an integer over 10^Z-1. All numbers not divisible by 2 or 5 share this property, so thus, all other numbers not already excluded (numbers divisible by 2, 3, or 5) are possible X values for part 3.

Posted by Gamer on 2006-10-16 14:59:54