Patterns with 1/13

 

1/13 = 0.076923076923076923076923076923076923076923076923076923076923...
0.0 \overline{769230}   (period 6)

2/13 = 0.153846153846153846153846153846153846153846153846153846153846...
0. \overline{153846}   (period 6)

3/13 = 0.230769230769230769230769230769230769230769230769230769230769...
0. \overline{230769}   (period 6)

4/13 = 0.307692307692307692307692307692307692307692307692307692307692...
0. \overline{307692}   (period 6)

5/13 = 0.384615384615384615384615384615384615384615384615384615384615...
0. \overline{384615}   (period 6)

6/13 = 0.461538461538461538461538461538461538461538461538461538461538...
0. \overline{461538}   (period 6)

7/13 = 0.538461538461538461538461538461538461538461538461538461538461...
0. \overline{538461}   (period 6)

8/13 = 0.615384615384615384615384615384615384615384615384615384615384...
0. \overline{615384}   (period 6)

9/13 = 0.692307692307692307692307692307692307692307692307692307692307...
0. \overline{692307}   (period 6)

10/13 = 0.769230769230769230769230769230769230769230769230769230769230...
0. \overline{769230}   (period 6)

11/13 = 0.846153846153846153846153846153846153846153846153846153846153...
0. \overline{846153}   (period 6)

12/13 = 0.923076923076923076923076923076923076923076923076923076923076...
0. \overline{923076}   (period 6)

 

769230 \; = \; 10 \; \times \: \; 76923
307692 \; = \; 4 \; \times \: \; 76923
230769 \; = \; 3 \; \times \: \; 76923
923076 \; = \; 12 \; \times \: \; 76923
692307 \; = \; 9 \; \times \: \; 76923

615384 \; = \; 4 \; \times \: \; 153846
461538 \; = \; 3 \; \times \: \; 153846
846153 \; = \; 11/2 \; \times \: \; 153846
384615 \; = \; 5/2 \; \times \: \; 153846
538461 \; = \; 7/2 \; \times \: \; 153846

923076 \; = \; 4 \; \times \: \; 230769
692307 \; = \; 3 \; \times \: \; 230769
769230 \; = \; 10/3 \; \times \: \; 230769
307692 \; = \; 4/3 \; \times \: \; 230769

230769 \; = \; 3/4 \; \times \: \; 307692
923076 \; = \; 3 \; \times \: \; 307692
692307 \; = \; 9/4 \; \times \: \; 307692
769230 \; = \; 5/2 \; \times \: \; 307692

538461 \; = \; 7/5 \; \times \: \; 384615
153846 \; = \; 2/5 \; \times \: \; 384615
615384 \; = \; 8/5 \; \times \: \; 384615
461538 \; = \; 6/5 \; \times \: \; 384615
846153 \; = \; 11/5 \; \times \: \; 384615

846153 \; = \; 11/6 \; \times \: \; 461538
384615 \; = \; 5/6 \; \times \: \; 461538
538461 \; = \; 7/6 \; \times \: \; 461538
615384 \; = \; 4/3 \; \times \: \; 461538

 

923^2 \; - \; 76^2 \; = \; 846153 \; = \; 11 \; \times \: \; 76923
846^2 \; - \; 153^2 \; = \; 692307 \; = \; 9 \; \times \: \; 76923
769^2 \; - \; 230^2 \; = \;  538461 \; = \; 7 \; \times \: \; 76923
692^2 \; - \; 307^2 \; = \; 384615 \; = \; 5 \; \times \: \; 76923
615^2 \; - \; 384^2 \; = \; 230769 \; = \; 3 \; \times \: \; 76923
538^2 \; - \; 461^2 \; = \; 76923

 

76923 \; \times \: \; 13 \; \times \: \; 1 \; = \; 0999999
76923 \; \times \: \; 13 \; \times \: \; 2 \; = \; 1999998
76923 \; \times \: \; 13 \; \times \: \; 3 \; = \; 2999997
76923 \; \times \: \; 13 \; \times \: \; 4 \; = \; 3999996
76923 \; \times \: \; 13 \; \times \: \; 5 \; = \; 4999995
76923 \; \times \: \; 13 \; \times \: \; 6 \; = \; 5999994
76923 \; \times \: \; 13 \; \times \: \; 7 \; = \; 6999993
76923 \; \times \: \; 13 \; \times \: \; 8 \; = \; 7999992
76923 \; \times \: \; 13 \; \times \: \; 9 \; = \; 8999991

76923 \; \times \: \; 13 \; \times \: \; 10 \; = \; 9999990
76923 \; \times \: \; 13 \; \times \: \; 11 \; = \; 10999989
76923 \; \times \: \; 13 \; \times \: \; 12 \; = \; 11999988
76923 \; \times \: \; 13 \; \times \: \; 13 \; = \; 12999987

7 \; \times \: \; 13 \; + \; 9 \; = \; 100
76 \; \times \: \; 13 \; + \; 12 \; = \; 1000
769 \; \times \: \; 13 \; + \; 3 \; = \; 10000
7692 \; \times \: \; 13 \; + \; 4 \; = \; 100000
76923 \; \times \: \; 13 \; + \; 1 \; = \; 1000000

 

                                                             ——————————————
 

Egyptian fraction expansion :

1/13
2/13 \; = \; 1/7 \; + \; 1/91
3/13 \; = \; 1/5 \; + \; 1/33 \; + \; 1/2145
4/13 \; = \; 1/4 \; + \; 1/18 \; + \; 1/468
5/13 \; = \; 1/3 \; + \; 1/20 \; + \; 1/780
6/13 \; = \; 1/3 \; + \; 1/8 \; + \; 1/312
7/13 \; = \; 1/2 \; + \; 1/26
8/13 \; = \; 1/2 \; + \; 1/9 \; + \; 1/234
9/13 \; = \; 1/2 + 1/6 \; + \; 1/39
10/13 \; = \; 1/2 \; + \; 1/4 \; + \; 1/52
11/13 \; = \; 1/2 \; + \; 1/3 \; + \; 1/78
12/13 \; = \; 1/2 \; + \; 1/3 \; + \; 1/12 \; + \; 1/156

 
 
 
 
 

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About benvitalis

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

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