Patterns with 1/7

 
 

1/7 = 0.142857142857142857142857142857142857142857142857142857142857...

0. \overline{142857}   (period 6)

Move 7 :     714285 \; = \; 5 \; \times \: 142857

2/7 = 0.285714285714285714285714285714285714285714285714285714285714...

0. \overline{285714}   (period 6)

Move 4 :     428571 \; = \; 3/2 \; \times \: 285714

3/7 = 0.428571428571428571428571428571428571428571428571428571428571...

0. \overline{428571}   (period 6)

Move 1 :     428571 \; = \; 3 \; \times \: 142857

4/7 = 0.571428571428571428571428571428571428571428571428571428571428...

0. \overline{571428}   (period 6)

Move 8 :     857142 \; = \; 3/2 \; \times \: 571428

5/7 = 0.714285714285714285714285714285714285714285714285714285714285...

0. \overline{714285}   (period 6)

Move 5 :     571428 \; = \; 4/5 \; \times \: 714285

6/7 = 0.857142857142857142857142857142857142857142857142857142857142...

0. \overline{857142}   (period 6)

Move 2 :     857142 \; = \; 3 \; \times \: 285714

 

Other rotations:

714285 \; = \; 5 \; \; \times \: \; 142857
571428 \; = \; 4 \; \; \times \: \; 142857
857142 \; = \; 6 \; \; \times \: \; 142857
285714 \; = \; 2 \; \; \times \: \; 142857
428571 \; = \; 3 \; \; \times \: \; 142857

 

142857,   285714,   428571,   571428,   714285,   857142 :

142857 \; \times \: \; 5 \; = \; 714285
714285 = 857^2 - 142^2
  where   857142 = 142857 \; \times \: \; 6

714^2 \; - \; 285^2 \; = \; 428571   where   428571 = 3 \; \times \: \; 142857

571^2 \; - \; 428^2 \; = \; 142857   where   571428 = 4 \; \times \: \; 142857

142857^2 = 20408122449 = 20408||122449 = (20408 + 122449)^2
285714^2 = 81632489796 = 81632||489796          81632+489796 = 571428 = 4 \; \times \: 142857
428571^2 = 183673102041 = 183673||102041      183673+102041 = 285714 = 2 \; \times \: 142857
571428^2 = 326529959184 = 326529||959184      326529+959184 = 1285713 = 9 \; \times \: 142857
714285^2 = 510203061225 = 510203||061225      510203+061225 = 571428 = 4 \; \times \: 142857
857142^2 = 734692408164 = 734692||408164      734692+408164 = 1142856 = 8 \; \times \: 142857

 
 

7000007 \; \times \: 142857 \; \times \: 1 \;  \; = \; 0999999999999
7000007 \; \times \: 142857 \; \times \: 2 \;  \; = \; 1999999999998
7000007 \; \times \: 142857 \; \times \: 3 \;  \; = \; 2999999999997
7000007 \; \times \: 142857 \; \times \: 4 \;  \; = \; 3999999999996
7000007 \; \times \: 142857 \; \times \: 5 \;  \; = \; 4999999999995
7000007 \; \times \: 142857 \; \times \: 6 \;  \; = \; 5999999999994
7000007 \; \times \: 142857 \; \times \: 7 \;  \; = \; 6999999999993
7000007 \; \times \: 142857 \; \times \: 8 \;  \; = \; 7999999999992
7000007 \; \times \: 142857 \; \times \: 9 \;  \; = \; 8999999999991
7000007 \; \times \: 142857 \; \times \: 10 \; \; = \; 9999999999990

7000007 \; \times \: 285714 \; \times \: 1 \; \; = \; \; 1999999999998
7000007 \; \times \: 285714 \; \times \: 2 \; \; = \; \; 3999999999996
7000007 \; \times \: 285714 \; \times \: 3 \; \; = \; \; 5999999999994
7000007 \; \times \: 285714 \; \times \: 4 \; \; = \; \; 7999999999992
7000007 \; \times \: 285714 \; \times \: 5 \; \; = \; \; 9999999999990
7000007 \; \times \: 285714 \; \times \: 6 \; \; = \; \; 11999999999988
7000007 \; \times \: 285714 \; \times \: 7 \; \; = \; \; 13999999999986
7000007 \; \times \: 285714 \; \times \: 8 \; \; = \; \; 15999999999984
7000007 \; \times \: 285714 \; \times \: 9 \; \; = \; \; 17999999999982
7000007 \; \times \: 285714 \; \times \: 10 \; \; = \; \; 19999999999980

7000007 \; \times \: \; 428571 \; \times \: \; 1 \; \; = \; \; 2999999999997
7000007 \; \times \: \; 428571 \; \times \: \; 2 \; \; = \; \; 5999999999994
7000007 \; \times \: \; 428571 \; \times \: \; 3 \; \; = \; \; 8999999999991
7000007 \; \times \: \; 428571 \; \times \: \; 4 \; \; = \; \; 11999999999988
7000007 \; \times \: \; 428571 \; \times \: \; 5 \; \; = \; \; 14999999999985
7000007 \; \times \: \; 428571 \; \times \: \; 6 \; \; = \; \; 17999999999982
7000007 \; \times \: \; 428571 \; \times \: \; 7 \; \; = \; \; 20999999999979
7000007 \; \times \: \; 428571 \; \times \: \; 8 \; \; = \; \; 23999999999976
7000007 \; \times \: \; 428571 \; \times \: \; 9 \; \; = \; \; 26999999999973
7000007 \; \times \: \; 428571 \; \times \: \; 10 \; \; = \; \; 29999999999970

7000007 \; \times \: \; 571428 \; \times \: \; 1 \; \; = \; \; 3999999999996
7000007 \; \times \: \; 571428 \; \times \: \; 2 \; \; = \; \; 7999999999992
7000007 \; \times \: \; 571428 \; \times \: \; 3 \; \; = \; \; 11999999999988
7000007 \; \times \: \; 571428 \; \times \: \; 4 \; \; = \; \; 15999999999984
7000007 \; \times \: \; 571428 \; \times \: \; 5 \; \; = \; \; 19999999999980
7000007 \; \times \: \; 571428 \; \times \: \; 6 \; \; = \; \; 23999999999976
7000007 \; \times \: \; 571428 \; \times \: \; 7 \; \; = \; \; 27999999999972
7000007 \; \times \: \; 571428 \; \times \: \; 8 \; \; = \; \; 31999999999968
7000007 \; \times \: \; 571428 \; \times \: \; 9 \; \; = \; \; 35999999999964
7000007 \; \times \: \; 571428 \; \times \: \; 10 \; \; = \; \; 39999999999960

7000007 \; \times \: \; 714285 \; \times \: \; 1 \; \; = \; \; 4999999999995
7000007 \; \times \: \; 714285 \; \times \: \; 2 \; \; = \; \; 9999999999990
7000007 \; \times \: \; 714285 \; \times \: \; 3 \; \; = \; \; 14999999999985
7000007 \; \times \: \; 714285 \; \times \: \; 4 \; \; = \; \; 19999999999980
7000007 \; \times \: \; 714285 \; \times \: \; 5 \; \; = \; \; 24999999999975
7000007 \; \times \: \; 714285 \; \times \: \; 6 \; \; = \; \; 29999999999970
7000007 \; \times \: \; 714285 \; \times \: \; 7 \; \; = \; \; 34999999999965
7000007 \; \times \: \; 714285 \; \times \: \; 8 \; \; = \; \; 39999999999960
7000007 \; \times \: \; 714285 \; \times \: \; 9 \; \; = \; \; 44999999999955
7000007 \; \times \: \; 714285 \; \times \: \; 10 \; \; = \; \; 49999999999950

7000007 \; \times \: \; 857142 \; \times \: \; 1 \; \; = \; \; 5999999999994
7000007 \; \times \: \; 857142 \; \times \: \; 2 \; \; = \; \; 11999999999988
7000007 \; \times \: \; 857142 \; \times \: \; 3 \; \; = \; \; 17999999999982
7000007 \; \times \: \; 857142 \; \times \: \; 4 \; \; = \; \; 23999999999976
7000007 \; \times \: \; 857142 \; \times \: \; 5 \; \; = \; \; 29999999999970
7000007 \; \times \: \; 857142 \; \times \: \; 6 \; \; = \; \; 35999999999964
7000007 \; \times \: \; 857142 \; \times \: \; 7 \; \; = \; \; 41999999999958
7000007 \; \times \: \; 857142 \; \times \: \; 8 \; \; = \; \; 47999999999952
7000007 \; \times \: \; 857142 \; \times \: \; 9 \; \; = \; \; 53999999999946
7000007 \; \times \: \; 857142 \; \times \: \; 10 \; \; = \; \; 59999999999940

 
 

1 \; \times \: \; 7 \; + \; 3 \; = \; 10
14 \; \times \: \; 7 \; + \; 2 \; = \; 100
142 \; \times \: \; 7 \; + \; 6 \; = \; 1000
1428 \; \times \: \; 7 \; + \; 4 \; = \; 10000
14285 \; \times \: \; 7 \; + \; 5 \; = \; 100000
142857 \; \times \: \; 7 \; + \; 1 \; = \; 1000000
1428571 \; \times \: \; 7 \; + \; 3 \; = \; 10000000
14285714 \; \times \: \; 7 \; + \; 2 \; = \; 100000000
142857142 \; \times \: \; 7 \; + \; 6 \; = \; 1000000000
1428571428 \; \times \: \; 7 \; + \; 4 \; = \; 10000000000
14285714285 \; \times \: \; 7 \; + \; 5 \; = \; 100000000000
142857142857 \; \times \: \; 7 \; + \; 1 \; = \; 1000000000000
1428571428571 \; \times \: \; 7 \; + \; 3 \; = \; 10000000000000
14285714285714 \; \times \: \; 7 \; + \; 2 \; = \; 100000000000000
142857142857142 \; \times \: \; 7 \; + \; 6 \; = \; 1000000000000000
1428571428571428 \; \times \: \; 7 \; + \; 4 \; = \; 10000000000000000
14285714285714285 \; \times \: \; 7 \; + \; 5 \; = \; 100000000000000000
142857142857142857 \; \times \: \; 7 \; + \; 1 \; = \; 1000000000000000000

2 \; \times \: \; 7 \; + \; 6 \; = \; 20
28 \; \times \: \; 7 \; + \; 4 \; = \; 200
285 \; \times \: \; 7 \; + \; 5 \; = \; 2000
2857 \; \times \: \; 7 \; + \; 1 \; = \; 20000
28571 \; \times \: \; 7 \; + \; 3 \; = \; 200000
285714 \; \times \: \; 7 \; + \; 2 \; = \; 2000000
2857142 \; \times \: \; 7 \; + \; 6 \; = \; 20000000
28571428 \; \times \: \; 7 \; + \; 4 \; = \; 200000000
285714285 \; \times \: \; 7 \; + \; 5 \; = \; 2000000000
2857142857 \; \times \: \; 7 \; + \; 1 \; = \; 20000000000
28571428571 \; \times \: \; 7 \; + \; 3 \; = \; 200000000000
285714285714 \; \times \: \; 7 \; + \; 2 \; = \; 2000000000000
……………………..

4 \; \times \: \; 7 \; + \; 2 \; = \; 30
42 \; \times \: \; 7 \; + \; 6 \; = \; 300
428 \; \times \: \; 7 \; + \; 4 \; = \; 3000
4285 \; \times \: \; 7 \; + \; 5 \; = \; 30000
42857 \; \times \: \; 7 \; + \; 1 \; = \; 300000
428571 \; \times \: \; 7 \; + \; 3 \; = \; 3000000
………………………..

5 \; \times \: \; 7 \; + \; 5 \; = \; 40
57 \; \times \: \; 7 \; + \; 1 \; = \; 400
571 \; \times \: \; 7 \; + \; 3 \; = \; 4000
5714 \; \times \: \; 7 \; + \; 2 \; = \; 40000
57142 \; \times \: \; 7 \; + \; 6 \; = \; 400000
571428 \; \times \: \; 7 \; + \; 4 \; = \; 4000000
……………………

7 \; \times \: \; 7 \; + \; 1 \; = \; 50
71 \; \times \: \; 7 \; + \; 3 \; = \; 500
714 \; \times \: \; 7 \; + \; 2 \; = \; 5000
7142 \; \times \: \; 7 \; + \; 6 \; = \; 50000
71428 \; \times \: \; 7 \; + \; 4 \; = \; 500000
714285 \; \times \: \; 7 \; + \; 5 \; = \; 5000000
……………………

8 \; \times \: \; 7 \; + \; 4 \; = \; 60
85 \; \times \: \; 7 \; + \; 5 \; = \; 600
857 \; \times \: \; 7 \; + \; 1 \; = \; 6000
8571 \; \times \: \; 7 \; + \; 3 \; = \; 60000
85714 \; \times \: \; 7 \; + \; 2 \; = \; 600000
857142 \; \times \: \; 7 \; + \; 6 \; = \; 6000000
…………………….

 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Uncategorized and tagged , , . Bookmark the permalink.

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