This is interesting... numbers 27 and 37, when you divide 10 by one of them, you have the other as a repeating decimal. Is there a name for this?
$$\begin{eqnarray} 10 / 27 &=& 0.370370370370\ldots\\ 10 / 37 &=& 0.270270270270\ldots \end{eqnarray} $$
This is an excellent thing to notice. Congratulations!
As far as I know there is no name for this specific phenomenon. To see why it happens, start with the observation that $$\frac1{999} = 0.001001001\ldots.$$
Then: $$\begin{eqnarray} \frac{27}{999} & = & 0.027027027\ldots \\ \frac{37}{999} & = & 0.037037037\ldots \\ \end{eqnarray}$$
Since $27\cdot 37 = 999$, the fractions on the left simplify to $\frac1{37}$ and $\frac1{27}$, respectively, and we have your observation.
We can observe the same thing happening with $3\cdot 333 = 999$:
$$\begin{eqnarray} \frac{3}{999} & = & \frac1{333} & = & 0.003003003\ldots \\ \frac{333}{999} & = & \frac13 & = & 0.333333333\ldots \\ \end{eqnarray}$$
or similarly, since $369\cdot 271 = 99999$ and $\frac1{99999} = 0.00001\;00001\;00001\ldots$:
$$\begin{eqnarray} \frac{271}{99999} & = & \frac1{369} & = & 0.00271\;00271\;00271\ldots \\ \frac{369}{99999} & = & \frac1{271} & = & 0.00369\;00369\;00369\ldots \\ \end{eqnarray}$$
The reason that $\frac1{999} = 0.001001001\ldots$ is because $0.001001001\ldots$ can be considered to be the sum of a geometric series: $$\frac1{1000} + \left(\frac1{1000}\right)^2 + \left(\frac1{1000}\right)^3 + \ldots.$$
When the sum of a series $x+x^2+x^3 \ldots$ exists, it is always $$\frac x{1-x}.$$ In this case we have $x=\frac1{1000}$, so the sum is $$\frac{\frac1{1000}}{1-\frac1{1000}} = \frac1{999}.$$ Other repeating decimals can be handled similarly.