Reciprocal of 37
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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 $999 = 27 \times 37$.
$\blacksquare$