Reciprocal of 97
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Theorem
The decimal expansion of the reciprocal of $97$ is as follows:
- $\dfrac 1 {97} = 0 \cdotp \dot 01030 \, 92783 \, 50515 \, 46391 \, 75257 \, 73195 \, 87628 \, 86597 \, 93814 \, 43298 \, 96907 \, 21649 \, 48453 \, 60824 \, 74226 \, 80412 \, 37113 \, 40206 \, 18556 \, \dot 7$
This sequence is A021101 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
Performing the calculation using long division:
0.01030927835051546391752577319587628865979381443298969072164948453608247422680412371134020618556701...
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97)1.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
97 485 873 679 582 679 291 679 776 388 388 291 485
-- --- --- --- --- --- --- --- --- --- --- --- ---
300 500 170 310 280 910 290 210 440 720 120 390 650
291 485 97 291 194 873 194 194 388 679 97 388 582
--- --- --- --- --- --- --- --- --- --- --- --- ---
900 150 730 190 860 370 960 160 520 410 230 200 680
873 97 679 97 776 291 983 97 485 388 194 194 679
--- --- --- --- --- --- --- --- --- --- --- --- ---
270 530 510 930 840 790 870 630 350 220 360 600 100
194 485 485 873 776 776 776 582 291 194 291 582 97
--- --- --- --- --- --- --- --- --- --- --- --- ---
760 450 250 570 640 140 940 480 590 260 690 180 ...
679 388 194 485 582 97 873 388 582 194 679 97
--- --- --- --- --- --- --- --- --- --- --- ---
810 620 560 850 580 430 670 920 800 660 110 830
776 582 485 776 485 388 582 873 776 582 97 776
--- --- --- --- --- --- --- --- --- --- --- ---
340 380 750 740 950 420 880 470 240 780 130 540
291 291 679 679 873 388 873 388 194 776 97 485
--- --- --- --- --- --- --- --- --- --- --- ---
490 890 710 610 770 320 700 820 460 400 330 550
485 873 679 582 679 291 679 776 388 388 291 485
$\blacksquare$
Historical Note
The decimal expansion of the reciprocal was famously known by heart by Alexander Craig Aitken.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $97$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $97$