# 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 $5$:

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

## 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.