 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  Find Friendly Fractions! (Posted on 2004-11-24) Can you explain the relationship between 27 and 37 that produces 1/27=0.037037037... and 1/37=0.027027027...? (That is, each number forms the other number's repeating decimal.)

Can you provide other similar examples, possibly with more or fewer digits?

 See The Solution Submitted by Old Original Oskar! Rating: 3.5455 (11 votes) Comments: ( Back to comment list | You must be logged in to post comments.) more fun and a reason | Comment 6 of 14 | Note that 27*37 = 999
1000/27 = 37+1/27 = 37.037037...
1000 = 1mod(27)  and also =1mod(37)
The situation will hold true whenever you have two numbers N and M such that:
10^k is 1 both mod(N) and mod(M)
or  N*M = 10^k - 1

9999=9*1111
1/9 = .11111111111...
1/1111 = .000900090009....

9999=99*101
1/99 = .01010101...
1/101 = .009900990099....

Trivial example:  9=3*3
1/3 = .33333333...

99999=271*369
1/271 = .003690036900369...
1/369 = .002710027100271...

 Posted by Larry on 2004-11-24 14:56:15 Please log in:

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