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Three Periodical Questions (Posted on 2006-10-16) Difficulty: 4 of 5
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)

Comments: ( Back to comment list | You must be logged in to post comments.)
Hints/Tips re: What is 'pure'? (and part 1 & 2)) | Comment 2 of 4 |
(In reply to What is 'pure'? (and part 1 & 2)) by Jer)

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.
  Posted by Federico Kereki on 2006-10-16 11:47:34

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