41 is Smallest Number whose Period of Reciprocal is 5

Theorem

$41$ is the first positive integer the decimal expansion of whose reciprocal has a period of $5$:

$\dfrac 1 {41} = 0 \cdotp \dot 0243 \dot 9$

This sequence is A021045 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Proof

From Reciprocal of $41$:

$\dfrac 1 {41} = 0 \cdotp \dot 0243 \dot 9$

Counting the digits, it is seen that this has a period of recurrence of $5$.

It remains to be shown that $41$ is the smallest positive integer which has this property.

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