Reciprocal of 67
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Theorem
The decimal expansion of the reciprocal of $67$ is as follows:
- $\dfrac 1 {67} = 0 \cdotp \dot 01492 \, 53731 \, 34328 \, 35820 \, 89552 \, 23880 \, 59 \dot 7$
This sequence is A021071 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
Performing the calculation using long division:
0.01492537313432835820895522388059701... ----------------------------------------- 67)1.00000000000000000000000000000000000... 67 ---- 330 210 560 640 590 268 201 536 603 536 --- --- --- --- --- 620 90 240 370 540 603 67 210 335 536 --- -- --- --- --- 170 230 390 350 400 134 201 335 335 335 --- --- --- --- --- 360 290 550 150 650 335 268 536 134 603 --- --- --- --- --- 250 220 140 160 470 201 201 134 134 469 --- --- --- --- --- 490 190 600 260 100 469 134 536 201 67 --- --- --- --- --- 210 560 640 590 ...
$\blacksquare$