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.
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?
(In reply to What is 'pure'? (and part 1 & 2))
Two things: you got the "pure" concept right, and in the Part 1 solution, you forgot to consider the (very easy) case when you get a zero remainder.