Theorem: Reciprocal of 131

:$\dfrac 1 {131} = 0 \cdotp \dot 00763 \, 35877 \, 86259 \, 54198 \, 47328 \, 24427 \, 48091 \, 60305 \, 34351 \, 14503 \, 81679 \, 38931 \, 29770 \, 99236 \, 64122 \, 13740 \, 45801 \, 52671 \, 75572 \, 51908 \, 39694 \, 65648 \, 85496 \, 18320 \, 61068 \, 7022 \dot 9$

Proof:
Performing the calculation using long division (zoom out for best effect):
<pre>
    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
</pre>
{{qed}}
Category:131
Category:Examples of Reciprocals
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