73 is Smallest Number whose Period of Reciprocal is 8

Theorem

$73$ is the first positive integer the decimal expansion of whose reciprocal has a period of $8$:

$\dfrac 1 {73} = 0 \cdotp \dot 01369 \, 86 \dot 3$

Proof

From Reciprocal of $73$:

$\dfrac 1 {73} = 0 \cdotp \dot 01369 \, 86 \dot 3$

Counting the digits, it is seen that this has a period of recurrence of $8$.

It remains to be shown that $73$ is the smallest positive integer which has this property.

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