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 Three Periodical Questions (Posted on 2006-10-16)
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?

 See The Solution Submitted by Federico Kereki Rating: 3.0000 (1 votes)

 Subject Author Date Solving part 3 Gamer 2006-10-16 14:59:54 Proof for part 2 Gamer 2006-10-16 13:43:46 re: What is 'pure'? (and part 1 & 2)) Federico Kereki 2006-10-16 11:47:34 What is 'pure'? (and part 1 & 2)) Jer 2006-10-16 11:09:06

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