Sweet Italian Girl and the Number 13

Peggy Sue’s Fraction:

Peggy Sue is a sweet Italian girl who has a thing about the number 13.  She found that the number 76,923,076,922,923,076,923,077 has an interesting property that relates to the number 13.  It is divisible by 13, but that’s not the amazing thing.  If you take the inverse of this number and convert it to a decimal expansion you get the 13 times table, starting with 0 * 13, and written in 12 digit chunks.

1 / 76,923,076,922,923,076,923,077 =

0.
000000000000  000000000013  000000000026  000000000039
000000000052  000000000065  000000000078  000000000091
000000000104  000000000117  000000000130  000000000143
000000000156  000000000169  000000000182  000000000195
000000000208  000000000221  000000000234  000000000247
000000000260  000000000273  000000000286  000000000299
000000000312  000000000325  000000000338  000000000351
000000000364  000000000377  000000000390  000000000403
000000000416  000000000429  000000000442  000000000455
000000000468  000000000481  000000000494  000000000507
000000000520  000000000533  000000000546  000000000559
000000000572  000000000585  000000000598  000000000611
000000000624  000000000637  000000000650  000000000663
000000000676  000000000689  000000000702  000000000715
000000000728  000000000741  000000000754  000000000767
000000000780  000000000793  000000000806  000000000819
000000000832  000000000845  000000000858  000000000871
000000000884  000000000897  000000000910  000000000923
000000000936  000000000949  000000000962  000000000975
000000000988  000000001001  000000001014  000000001027
000000001040  000000001053  000000001066  000000001079
000000001092  000000001105  000000001118  000000001131
000000001144  000000001157  000000001170  000000001183
000000001196  000000001209  000000001222  000000001235
000000001248  000000001261  000000001274  000000001287
000000001300  000000001313  000000001326  000000001339
000000001352  000000001365  000000001378  000000001391
000000001404  000000001417  000000001430  000000001443
000000001456  000000001469  000000001482  000000001495
000000001508  000000001521  000000001534  000000001547
000000001560  000000001573  000000001586  000000001599
000000001612  000000001625  000000001638  000000001651
000000001664  000000001677  000000001690  000000001703
000000001716  000000001729  000000001742  000000001755
000000001768  000000001781  000000001794  000000001807
000000001820  000000001833  000000001846  000000001859
000000001872  000000001885  000000001898  000000001911
000000001924  000000001937  000000001950  000000001963
000000001976  000000001989  000000002002  000000002015
000000002028  000000002041  000000002054  000000002067
000000002080  000000002093  000000002106  000000002119
000000002132  000000002145  000000002158  000000002171
000000002184  000000002197  000000002210  000000002223
000000002236  000000002249  000000002262  000000002275
000000002288  000000002301  000000002314  000000002327
000000002340  000000002353  000000002366  000000002379
000000002392  000000002405  000000002418  000000002431
000000002444  000000002457  000000002470  000000002483
000000002496  000000002509  000000002522  000000002535
000000002548  000000002561  000000002574  000000002587
000000002600  000000002613  000000002626  000000002639
000000002652  000000002665  000000002678  000000002691
000000002704  000000002717  000000002730  000000002743
000000002756  000000002769  000000002782  000000002795
000000002808  000000002821  000000002834  000000002847
000000002860  000000002873  000000002886  000000002899
000000002912  000000002925  000000002938  000000002951
000000002964  000000002977  000000002990  000000003003
000000003016  000000003029  000000003042  000000003055
000000003068  000000003081  000000003094  000000003107
000000003120  000000003133  000000003146  000000003159
000000003172  000000003185  000000003198  000000003211
000000003224  000000003237  000000003250  000000003263
000000003276  000000003289  000000003302  000000003315
000000003328  000000003341  000000003354  000000003367
000000003380  000000003393  000000003406  000000003419
000000003432  000000003445  000000003458  000000003471
000000003484  000000003497  000000003510  000000003523
000000003536  000000003549  000000003562  000000003575
000000003588  000000003601  000000003614  000000003627
000000003640  000000003653  000000003666  000000003679
000000003692  000000003705  000000003718  000000003731
000000003744  000000003757  000000003770  000000003783
000000003796  000000003809  000000003822  000000003835
000000003848  000000003861  000000003874  000000003887
000000003900  000000003913  000000003926  000000003939
000000003952  000000003965  000000003978  000000003991
000000004004  000000004017  000000004030  000000004043
000000004056  000000004069  000000004082  000000004095
000000004108  000000004121  000000004134  000000004147
000000004160  000000004173  000000004186  000000004199
000000004212  000000004225  ...

How long does this pattern continue?  I don’t know yet.  It takes time to do long division.

Back to Fibonacci review tomorrow.

David