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?

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