101 is Smallest Number whose Period of Reciprocal is 4

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Theorem

$101$ is the first positive integer the decimal expansion of whose reciprocal has a period of $4$:

$\dfrac 1 {101} = 0 \cdotp \dot 009 \dot 9$

This sequence is A021105 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Proof

From Reciprocal of $101$:

$\dfrac 1 {101} = 0 \cdotp \dot 009 \dot 9$

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