Reciprocal of 131
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Theorem
The decimal expansion of the reciprocal of $131$ is as follows:
- $\dfrac 1 {131} = 0 \cdotp \dot 00763 \, 35877 \, 86259 \, 54198 \, 47328 \, 24427 \, 48091 \, 60305 \, 34351 \, 14503 \, 81679 \, 38931 \, 29770 \, 99236 \, 64122 \, 13740 \, 45801 \, 52671 \, 75572 \, 51908 \, 39694 \, 65648 \, 85496 \, 18320 \, 61068 \, 7022 \dot 9$
This sequence is A021135 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
Performing the calculation using long division (zoom out for best effect):
0.0076335877862595419847328244274809160305343511450381679389312977099236641221374045801526717557251908396946564885496183206106870229
--------------------------------------------------------------------------------------------------------------------------------------
131)1.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
917 655 262 131 393 131 786 1179 786
--- --- --- --- --- --- --- ---- ---
830 550 980 190 170 290 940 910 240
786 524 917 131 131 262 917 786 131
--- --- --- --- --- --- --- --- ---
440 260 630 590 390 280 230 1240 1090
393 131 524 524 262 262 131 1179 1048
--- --- --- --- --- --- --- ---- ----
470 1290 1060 660 1280 180 990 610 420
393 1179 1048 655 1179 131 917 524 393
--- ---- ---- --- ---- --- --- --- ---
770 1110 1200 500 1010 490 730 860 270
655 1048 1179 393 917 393 655 786 262
--- ---- ---- --- --- --- --- --- ---
1150 620 210 1070 930 970 750 740 800
1048 524 131 1048 917 917 655 655 786
---- --- --- ---- --- --- --- --- ---
1020 960 790 220 1300 530 950 850 140
917 917 786 131 1179 524 917 786 131
---- --- --- --- ---- --- --- --- ---
1030 430 400 890 1210 600 330 640 900
917 393 393 786 1179 524 262 524 786
---- --- --- --- ---- --- --- --- ---
1130 370 700 1040 310 760 680 1160 1140
1048 262 655 917 262 655 655 1048 1048
---- --- --- ---- --- --- --- ---- ----
820 1080 450 1230 480 1050 250 1120 920
786 1048 393 1179 393 1048 131 1048 917
--- ---- --- ---- --- ---- --- ---- ---
340 320 570 510 870 200 1190 720 300
262 262 524 393 786 131 1179 655 262
--- --- --- --- --- --- ---- --- ---
780 580 460 1170 840 690 1100 650 380
655 524 393 1048 786 655 1048 524 262
--- --- --- ---- --- --- ---- --- ---
1250 560 670 1220 540 350 520 1260 1180
1179 524 655 1179 524 262 393 1179 1179
---- --- --- ---- --- --- --- ---- ----
710 360 150 410 160 880 1270 810 1000
655 262 131 393 131 786 1179 786 ....
--- --- --- --- --- --- ---- ---
550 980 190 170 290 940 910 240
$\blacksquare$