Reciprocal of 37

Theorem

The decimal expansion of the reciprocal of $37$ is as follows:

$\dfrac 1 {37} = 0 \cdotp \dot 02 \dot 7$

This sequence is A021041 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Proof

Performing the calculation using long division:

   0.027...
  --------
37)1.00000
     74
     --
     260
     259
     ---
       100
        74
       ---
        ...

It is to be noted that this is because $999 = 27 \times 37$.

$\blacksquare$