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?
Say 1/X=0.YYY.... If Y is n digits long, then the fraction equals Y times 10-n+10-2n+10-3n...= Y/(10n-1). Therefore, we need XY=10n-1. In our case, 27x37=103-1; other examples are 3x3=101-1 (1/3=0.333...) or 11x9=102-1 (1/11=0.090909...) and so on.
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