Moving one digit from one end to the other of a number

 
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

 

2/13 = 0.153846153846153846153846153846153846153846153846153846153846...

0. \overline{153846}   (period 6)

Move 6 :
615384 \; = \; 4 \; \times \: 153846

 

4/39 = 0.102564102564102564102564102564102564102564102564102564102564...

0. \overline{102564}   (period 6)

Move 4 :
410256 \; = \; 4 \; \times \: 102564

5/39 = 0.128205128205128205128205128205128205128205128205128205128205...

0. \overline{128205}   (period 6)

Move 5 :
512820 \; = \; 4 \; \times \: 128205

7/39 = 0.179487179487179487179487179487179487179487179487179487179487...

0. \overline{179487}   (period 6)

Move 7 :
717948 \; = \; 4 \; \times \: 179487

 

8/79 = 0.101265822784810126582278481012658227848101265822784810126582...

0. \overline{1012658227848}   (period 13)

move 8 :
8101265822784 \; = \; 8 \; \times \: 1012658227848

9/79 = 0.113924050632911392405063291139240506329113924050632911392405...

0. \overline{1139240506329}   (period 13)

move 9 :
9113924050632 \; = \; 8 \; \times \: 1139240506329

 

7/19 = 0.368421052631578947368421052631578947368421052631578947368421...

0. \overline{368421052631578947}   (period 18)

Move 7 :
736842105263157894 \; = \; 2 \; \times \: 368421052631578947

9/19 = 0.473684210526315789473684210526315789473684210526315789473684...

0.\overline{473684210526315789}   (period 18)

move 9 :
947368421052631578 \; = \; 2 \; \times \: 473684210526315789

 

3/23 = 0.130434782608695652173913043478260869565217391304347826086956...

0. \overline{1304347826086956521739}   (period 22)

Move 9 :
9130434782608695652173 \; = \; 7 \; \times \: 1304347826086956521739

 

9/29 = 0.310344827586206896551724137931034482758620689655172413793103...

0. \overline{3103448275862068965517241379}   (period 28)

Move 9 :
9310344827586206896551724137 \; = \;  3 \; \times \: 3103448275862068965517241379

 

9/49 = 0.183673469387755102040816326530612244897959183673469387755102...

0. \overline{183673469387755102040816326530612244897959}   (period 42)

Move 9 :
918367346938775510204081632653061224489795 = 5 \times \: 183673469387755102040816326530612244897959

 

9/89 = 0.101123595505617977528089887640449438202247191011235955056179...

0. \overline{10112359550561797752808988764044943820224719}   (period 44)

Move 9 :
10112359550561797752808988764044943820224719
\; \times \: 9
= \; 91011235955056179775280898876404494382022471

 

9/59 = 0.152542372881355932203389830508474576271186440677966101694915...

0. \overline{1525423728813559322033898305084745762711864406779661016949}   (period 58)

Move 9 :

1525423728813559322033898305084745762711864406779661016949
\; \times \: 6
= \; 9152542372881355932203389830508474576271186440677966101694

 
Find other examples.

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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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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