Period of Reciprocal of 27 is Smallest with Length 3
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Theorem
$27$ is the smallest positive integer the decimal expansion of whose reciprocal has a period of $3$:
- $\dfrac 1 {27} = 0 \cdotp \dot 03 \dot 7$
This sequence is A021027 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
Performing the calculation using long division:
0.037... -------- 27)1.00000 81 -- 190 189 --- 100 81 --- ...
This is because $999 = 27 \times 37$.
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): $999$