There exists a interesting relationship between \(27\) and \(37\)

\(\frac{1}{27}=0.037037037...\) and \(\frac{1}{37}=0.027027027...\) (That is, each number forms the other number's repeating decimal.)

This can be explained as follows :

Say \(\frac{1}{X}=0.YYY....\) If \(Y\) is n digits long,

then the fraction equals \(Y\) times \(10^{-n}+10^{-2n}+10^{-3n}...= \frac{Y}{(10^n-1)}\).

Therefore, we need \(XY=10^n-1\). In the above case, \(27*37=10^3-1\); other examples are \(3*3=10^1-1\)

So \(\frac{1}{3}=0.333...\)

or \(11*9=10^2-1\) so \(\frac{1}{11}=0.090909...\) and \(\frac{1}{9}=0.11111...\) and so on.

## Comments

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TopNewestOh that's nice! Thanks for sharing this pattern that you noticed. – Calvin Lin Staff · 2 years, 4 months ago

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!!! – Alfian Edgar · 2 years, 4 months ago

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It is Fantastic! Thanks for sharing @Danish Ahmed – Sarthak Tanwani · 2 years, 4 months ago

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nice – Akash Gupta · 2 years, 4 months ago

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Excellent – Kanthi Deep · 2 years, 4 months ago

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