27 is Smallest Number whose Period of Reciprocal is 3

Theorem

$27$ is the smallest positive integer the decimal expansion of whose reciprocal has a period of $3$:

$\dfrac 1 {27} = 0 \cdotp \dot 03 \dot 7$

Proof

From Reciprocal of $27$:

$\dfrac 1 {27} = 0 \cdotp \dot 03 \dot 7$

It can be determined by inspection of all smaller integers that this is indeed the smallest to have a period of $3$.

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $27$
• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $999$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $27$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $999$