2025 162 81 R O1 O2

n/2025

Obsérvense los periodos en cada columna

En la última fila vemos algunas observaciones

\[ \frac{1}{2025} = 0.00\overline{\color{#000080}{\mathbf{049382716}}} \] \[ \frac{2}{2025} = 0.00\overline{\color{#000080}{\mathbf{098765432}}} \] \[ \frac{3}{2025} = 0.00\overline{\color{#000080}{\mathbf{148}}} \] \[ \frac{4}{2025} = 0.00\overline{\color{#000080}{\mathbf{197530864}}} \] \[ \frac{5}{2025} = 0.00\overline{\color{#000080}{\mathbf{246913580}}} \] \[ \frac{6}{2025} = 0.00\overline{\color{#000080}{\mathbf{296}}} \] \[ \frac{7}{2025} = 0.00\overline{\color{#000080}{\mathbf{345679012}}} \] \[ \frac{8}{2025} = 0.00\overline{\color{#000080}{\mathbf{395061728}}} \] \[ \frac{9}{2025} = 0.00\overline{\color{#000080}{\mathbf{4}}} \]
\[ \frac{10}{2025} = 0.00\overline{\color{#000080}{\mathbf{493827160}}} \] \[ \frac{11}{2025} = 0.00\overline{\color{#000080}{\mathbf{543209876}}} \] \[ \frac{12}{2025} = 0.00\overline{\color{#000080}{\mathbf{592}}} \] \[ \frac{13}{2025} = 0.00\overline{\color{#000080}{\mathbf{641975308}}} \] \[ \frac{14}{2025} = 0.00\overline{\color{#000080}{\mathbf{691358024}}} \] \[ \frac{15}{2025} = 0.00\overline{\color{#000080}{\mathbf{740}}} \] \[ \frac{16}{2025} = 0.00\overline{\color{#000080}{\mathbf{790123456}}} \] \[ \frac{17}{2025} = 0.00\overline{\color{#000080}{\mathbf{839506172}}} \] \[ \frac{18}{2025} = 0.00\overline{\color{#000080}{\mathbf{8}}} \]
\[ \frac{19}{2025} = 0.00\overline{\color{#000080}{\mathbf{938271604}}} \] \[ \frac{20}{2025} = 0.00\overline{\color{#000080}{\mathbf{987654320}}} \] \[ \frac{21}{2025} = 0.01\overline{\color{#000080}{\mathbf{037}}} \] \[ \frac{22}{2025} = 0.01\overline{\color{#000080}{\mathbf{086419753}}} \] \[ \frac{23}{2025} = 0.01\overline{\color{#000080}{\mathbf{135802469}}} \] \[ \frac{24}{2025} = 0.01\overline{\color{#000080}{\mathbf{185}}} \] \[ \frac{25}{2025} = 0.0\overline{\color{#000080}{\mathbf{123456790}}} \] \[ \frac{26}{2025} = 0.01\overline{\color{#000080}{\mathbf{283950617}}} \] \[ \frac{27}{2025} = 0.01\overline{\color{#000080}{\mathbf{3}}} \]
\[ \frac{28}{2025} = 0.01\overline{\color{#000080}{\mathbf{382716049}}} \] \[ \frac{29}{2025} = 0.01\overline{\color{#000080}{\mathbf{432098765}}} \] \[ \frac{30}{2025} = 0.01\overline{\color{#000080}{\mathbf{481}}} \] \[ \frac{31}{2025} = 0.01\overline{\color{#000080}{\mathbf{530864197}}} \] \[ \frac{32}{2025} = 0.01\overline{\color{#000080}{\mathbf{580246913}}} \] \[ \frac{33}{2025} = 0.01\overline{\color{#000080}{\mathbf{629}}} \] \[ \frac{34}{2025} = 0.01\overline{\color{#000080}{\mathbf{679012345}}} \] \[ \frac{35}{2025} = 0.01\overline{\color{#000080}{\mathbf{728395061}}} \] \[ \frac{36}{2025} = 0.01\overline{\color{#000080}{\mathbf{7}}} \]
\[ \frac{37}{2025} = 0.01\overline{\color{#000080}{\mathbf{827160493}}} \] \[ \frac{38}{2025} = 0.01\overline{\color{#000080}{\mathbf{876543209}}} \] \[ \frac{39}{2025} = 0.01\overline{\color{#000080}{\mathbf{925}}} \] \[ \frac{40}{2025} = 0.01\overline{\color{#000080}{\mathbf{975308641}}} \] \[ \frac{41}{2025} = 0.02\overline{\color{#000080}{\mathbf{024691358}}} \] \[ \frac{42}{2025} = 0.02\overline{\color{#000080}{\mathbf{074}}} \] \[ \frac{43}{2025} = 0.02\overline{\color{#000080}{\mathbf{12345679}}} \] \[ \frac{44}{2025} = 0.02\overline{\color{#000080}{\mathbf{172839506}}} \] \[ \frac{45}{2025} = 0.02\overline{\color{#000080}{\mathbf{2}}} \]
\[ \frac{46}{2025} = 0.02\overline{\color{#000080}{\mathbf{271604938}}} \] \[ \frac{47}{2025} = 0.02\overline{\color{#000080}{\mathbf{320987654}}} \] \[ \frac{48}{2025} = 0.02\overline{\color{#000080}{\mathbf{370}}} \] \[ \frac{49}{2025} = 0.02\overline{\color{#000080}{\mathbf{419753086}}} \] \[ \frac{50}{2025} = 0.02\overline{\color{#000080}{\mathbf{469135802}}} \] \[ \frac{51}{2025} = 0.02\overline{\color{#000080}{\mathbf{518}}} \] \[ \frac{52}{2025} = 0.02\overline{\color{#000080}{\mathbf{567901234}}} \] \[ \frac{53}{2025} = 0.02\overline{\color{#000080}{\mathbf{617283950}}} \] \[ \frac{54}{2025} = 0.02\overline{\color{#000080}{\mathbf{6}}} \]
\[ \frac{55}{2025} = 0.02\overline{\color{#000080}{\mathbf{716049382}}} \] \[ \frac{56}{2025} = 0.02\overline{\color{#000080}{\mathbf{765432098}}} \] \[ \frac{57}{2025} = 0.02\overline{\color{#000080}{\mathbf{814}}} \] \[ \frac{58}{2025} = 0.02\overline{\color{#000080}{\mathbf{864197530}}} \] \[ \frac{59}{2025} = 0.02\overline{\color{#000080}{\mathbf{913580246}}} \] \[ \frac{60}{2025} = 0.02\overline{\color{#000080}{\mathbf{962}}} \] \[ \frac{61}{2025} = 0.03\overline{\color{#000080}{\mathbf{012345679}}} \] \[ \frac{62}{2025} = 0.03\overline{\color{#000080}{\mathbf{061728395}}} \] \[ \frac{63}{2025} = 0.03\overline{\color{#000080}{\mathbf{1}}} \]
\[ \frac{64}{2025} = 0.03\overline{\color{#000080}{\mathbf{160493827}}} \] \[ \frac{65}{2025} = 0.03\overline{\color{#000080}{\mathbf{209876543}}} \] \[ \frac{66}{2025} = 0.03\overline{\color{#000080}{\mathbf{259}}} \] \[ \frac{67}{2025} = 0.03\overline{\color{#000080}{\mathbf{308641975}}} \] \[ \frac{68}{2025} = 0.03\overline{\color{#000080}{\mathbf{358024691}}} \] \[ \frac{69}{2025} = 0.03\overline{\color{#000080}{\mathbf{407}}} \] \[ \frac{70}{2025} = 0.03\overline{\color{#000080}{\mathbf{456790123}}} \] \[ \frac{71}{2025} = 0.03\overline{\color{#000080}{\mathbf{506172839}}} \] \[ \frac{72}{2025} = 0.03\overline{\color{#000080}{\mathbf{5}}} \]
\[ \frac{73}{2025} = 0.03\overline{\color{#000080}{\mathbf{604938271}}} \] \[ \frac{74}{2025} = 0.03\overline{\color{#000080}{\mathbf{654320987}}} \] \[ \frac{75}{2025} = 0.03\overline{\color{#000080}{\mathbf{703}}} \] \[ \frac{76}{2025} = 0.03\overline{\color{#000080}{\mathbf{753086419}}} \] \[ \frac{77}{2025} = 0.03\overline{\color{#000080}{\mathbf{802469135}}} \] \[ \frac{78}{2025} = 0.03\overline{\color{#000080}{\mathbf{851}}} \] \[ \frac{79}{2025} = 0.03\overline{\color{#000080}{\mathbf{901234567}}} \] \[ \frac{80}{2025} = 0.03\overline{\color{#000080}{\mathbf{950617283}}} \] \[ \frac{81}{2025} = 0.04\overline{\color{#000080}{\mathbf{0}}} \]
\[ \frac{82}{2025} = 0.04\overline{\color{#000080}{\mathbf{049382716}}} \] \[ \frac{83}{2025} = 0.04\overline{\color{#000080}{\mathbf{098765432}}} \] \[ \frac{84}{2025} = 0.04\overline{\color{#000080}{\mathbf{148}}} \] \[ \frac{85}{2025} = 0.04\overline{\color{#000080}{\mathbf{197530864}}} \] \[ \frac{86}{2025} = 0.04\overline{\color{#000080}{\mathbf{246913580}}} \] \[ \frac{87}{2025} = 0.04\overline{\color{#000080}{\mathbf{296}}} \] \[ \frac{88}{2025} = 0.04\overline{\color{#000080}{\mathbf{345679012}}} \] \[ \frac{89}{2025} = 0.04\overline{\color{#000080}{\mathbf{395061728}}} \] \[ \frac{90}{2025} = 0.04\overline{\color{#000080}{\mathbf{4}}} \]
\[ \frac{91}{2025} = 0.04\overline{\color{#000080}{\mathbf{493827160}}} \] \[ \frac{92}{2025} = 0.04\overline{\color{#000080}{\mathbf{543209876}}} \] \[ \frac{93}{2025} = 0.04\overline{\color{#000080}{\mathbf{592}}} \] \[ \frac{94}{2025} = 0.04\overline{\color{#000080}{\mathbf{641975308}}} \] \[ \frac{95}{2025} = 0.04\overline{\color{#000080}{\mathbf{691358024}}} \] \[ \frac{96}{2025} = 0.04\overline{\color{#000080}{\mathbf{740}}} \] \[ \frac{97}{2025} = 0.04\overline{\color{#000080}{\mathbf{790123456}}} \] \[ \frac{98}{2025} = 0.04\overline{\color{#000080}{\mathbf{839506172}}} \] \[ \frac{99}{2025} = 0.04\overline{\color{#000080}{\mathbf{8}}} \]
periodo 049382716

de 1 en 1

sin 5

 periodo 098765432

de 5 en 5

sin 1

 periodos 148, 592, 037

sin 6

periodo 197530864

de 7 en 7

sin 2

 periodo 024691358

de 2 en 2

sin 7

 periodo 296, 074, 185

sin 3

periodo 345679012

de 4 en 4

sin 8

 periodo 395061728

de 8 en 8

sin 4

 periodo 4,8,3,7,2,6,1,5,0,

sin 9

En la primera columna al periodo le falta el 5

En la 2ª columna al periodo le falta el 1 pues 2·5=10 y 1+0=1

En la 3ª columna al periodo le falta el 6 pues 3·5=15 y 1+5=6

En la 4ª columna al periodo le falta el 2 pues 4·5=20 y 2+0=2

...

En la 9ª columna al periodo le falta el 9 pues 9·5=45 y 4+5=9

n/162

Obsérvense los periodos en cada columna

En la última fila vemos algunas observaciones

\[ \frac{1}{162} = 0.0\overline{\color{#000080}{\mathbf{061728395}}} \] \[ \frac{2}{162} = 0.0\overline{\color{#000080}{\mathbf{123456790}}} \] \[ \frac{3}{162} = 0.0\overline{\color{#000080}{\mathbf{185}}} \] \[ \frac{4}{162} = 0.0\overline{\color{#000080}{\mathbf{246913580}}} \] \[ \frac{5}{162} = 0.0\overline{\color{#000080}{\mathbf{308641975}}} \] \[ \frac{6}{162} = 0.0\overline{\color{#000080}{\mathbf{370}}} \] \[ \frac{7}{162} = 0.0\overline{\color{#000080}{\mathbf{432098765}}} \] \[ \frac{8}{162} = 0.0\overline{\color{#000080}{\mathbf{493827160}}} \] \[ \frac{9}{162} = 0.0\overline{\color{#000080}{\mathbf{5}}} \]
\[ \frac{10}{162} = 0.0\overline{\color{#000080}{\mathbf{617283950}}} \] \[ \frac{11}{162} = 0.0\overline{\color{#000080}{\mathbf{679012345}}} \] \[ \frac{12}{162} = 0.0\overline{\color{#000080}{\mathbf{740}}} \] \[ \frac{13}{162} = 0.0\overline{\color{#000080}{\mathbf{802469135}}} \] \[ \frac{14}{162} = 0.0\overline{\color{#000080}{\mathbf{864197530}}} \] \[ \frac{15}{162} = 0.0\overline{\color{#000080}{\mathbf{925}}} \] \[ \frac{16}{162} = 0.0\overline{\color{#000080}{\mathbf{987654320}}} \] \[ \frac{17}{162} = 0.1\overline{\color{#000080}{\mathbf{049382716}}} \] \[ \frac{18}{162} = 0.1\overline{\color{#000080}{\mathbf{1}}} \]
\[ \frac{19}{162} = 0.1\overline{\color{#000080}{\mathbf{172839506}}} \] \[ \frac{20}{162} = 0.1\overline{\color{#000080}{\mathbf{234567901}}} \] \[ \frac{21}{162} = 0.1\overline{\color{#000080}{\mathbf{296}}} \] \[ \frac{22}{162} = 0.1\overline{\color{#000080}{\mathbf{358024691}}} \] \[ \frac{23}{162} = 0.1\overline{\color{#000080}{\mathbf{419753086}}} \] \[ \frac{24}{162} = 0.1\overline{\color{#000080}{\mathbf{481}}} \] \[ \frac{25}{162} = 0.1\overline{\color{#000080}{\mathbf{543209876}}} \] \[ \frac{26}{162} = 0.1\overline{\color{#000080}{\mathbf{604938271}}} \] \[ \frac{27}{162} = 0.1\overline{\color{#000080}{\mathbf{6}}} \]
\[ \frac{28}{162} = 0.1\overline{\color{#000080}{\mathbf{728395061}}} \] \[ \frac{29}{162} = 0.1\overline{\color{#000080}{\mathbf{790123456}}} \] \[ \frac{30}{162} = 0.1\overline{\color{#000080}{\mathbf{851}}} \] \[ \frac{31}{162} = 0.1\overline{\color{#000080}{\mathbf{913580246}}} \] \[ \frac{32}{162} = 0.1\overline{\color{#000080}{\mathbf{975308641}}} \] \[ \frac{33}{162} = 0.2\overline{\color{#000080}{\mathbf{037}}} \] \[ \frac{34}{162} = 0.2\overline{\color{#000080}{\mathbf{098765432}}} \] \[ \frac{35}{162} = 0.2\overline{\color{#000080}{\mathbf{160493827}}} \] \[ \frac{36}{162} = 0.2\overline{\color{#000080}{\mathbf{2}}} \]
\[ \frac{37}{162} = 0.2\overline{\color{#000080}{\mathbf{283950617}}} \] \[ \frac{38}{162} = 0.2\overline{\color{#000080}{\mathbf{345679012}}} \] \[ \frac{39}{162} = 0.2\overline{\color{#000080}{\mathbf{407}}} \] \[ \frac{40}{162} = 0.2\overline{\color{#000080}{\mathbf{469135802}}} \] \[ \frac{41}{162} = 0.2\overline{\color{#000080}{\mathbf{530864197}}} \] \[ \frac{42}{162} = 0.2\overline{\color{#000080}{\mathbf{592}}} \] \[ \frac{43}{162} = 0.2\overline{\color{#000080}{\mathbf{654320987}}} \] \[ \frac{44}{162} = 0.2\overline{\color{#000080}{\mathbf{716049382}}} \] \[ \frac{45}{162} = 0.2\overline{\color{#000080}{\mathbf{7}}} \]
\[ \frac{46}{162} = 0.2\overline{\color{#000080}{\mathbf{839506172}}} \] \[ \frac{47}{162} = 0.2\overline{\color{#000080}{\mathbf{901234567}}} \] \[ \frac{48}{162} = 0.2\overline{\color{#000080}{\mathbf{962}}} \] \[ \frac{49}{162} = 0.3\overline{\color{#000080}{\mathbf{024691358}}} \] \[ \frac{50}{162} = 0.3\overline{\color{#000080}{\mathbf{086419753}}} \] \[ \frac{51}{162} = 0.3\overline{\color{#000080}{\mathbf{148}}} \] \[ \frac{52}{162} = 0.3\overline{\color{#000080}{\mathbf{209876543}}} \] \[ \frac{53}{162} = 0.3\overline{\color{#000080}{\mathbf{271604938}}} \] \[ \frac{54}{162} = 0.3\overline{\color{#000080}{\mathbf{3}}} \]
\[ \frac{55}{162} = 0.3\overline{\color{#000080}{\mathbf{395061728}}} \] \[ \frac{56}{162} = 0.3\overline{\color{#000080}{\mathbf{456790123}}} \] \[ \frac{57}{162} = 0.3\overline{\color{#000080}{\mathbf{518}}} \] \[ \frac{58}{162} = 0.3\overline{\color{#000080}{\mathbf{580246913}}} \] \[ \frac{59}{162} = 0.3\overline{\color{#000080}{\mathbf{641975308}}} \] \[ \frac{60}{162} = 0.3\overline{\color{#000080}{\mathbf{703}}} \] \[ \frac{61}{162} = 0.3\overline{\color{#000080}{\mathbf{765432098}}} \] \[ \frac{62}{162} = 0.3\overline{\color{#000080}{\mathbf{827160493}}} \] \[ \frac{63}{162} = 0.3\overline{\color{#000080}{\mathbf{8}}} \]
\[ \frac{64}{162} = 0.3\overline{\color{#000080}{\mathbf{950617283}}} \] \[ \frac{65}{162} = 0.4\overline{\color{#000080}{\mathbf{012345679}}} \] \[ \frac{66}{162} = 0.4\overline{\color{#000080}{\mathbf{074}}} \] \[ \frac{67}{162} = 0.4\overline{\color{#000080}{\mathbf{135802469}}} \] \[ \frac{68}{162} = 0.4\overline{\color{#000080}{\mathbf{197530864}}} \] \[ \frac{69}{162} = 0.4\overline{\color{#000080}{\mathbf{259}}} \] \[ \frac{70}{162} = 0.4\overline{\color{#000080}{\mathbf{320987654}}} \] \[ \frac{71}{162} = 0.4\overline{\color{#000080}{\mathbf{382716049}}} \] \[ \frac{72}{162} = 0.4\overline{\color{#000080}{\mathbf{4}}} \]
\[ \frac{73}{162} = 0.4\overline{\color{#000080}{\mathbf{506172839}}} \] \[ \frac{74}{162} = 0.4\overline{\color{#000080}{\mathbf{567901234}}} \] \[ \frac{75}{162} = 0.4\overline{\color{#000080}{\mathbf{629}}} \] \[ \frac{76}{162} = 0.4\overline{\color{#000080}{\mathbf{691358024}}} \] \[ \frac{77}{162} = 0.4\overline{\color{#000080}{\mathbf{753086419}}} \] \[ \frac{78}{162} = 0.4\overline{\color{#000080}{\mathbf{814}}} \] \[ \frac{79}{162} = 0.4\overline{\color{#000080}{\mathbf{876543209}}} \] \[ \frac{80}{162} = 0.4\overline{\color{#000080}{\mathbf{938271604}}} \] \[ \frac{81}{162} = 0.5\overline{\color{#000080}{\mathbf{0}}} \]
\[ \frac{82}{162} = 0.5\overline{\color{#000080}{\mathbf{061728395}}} \] \[ \frac{83}{162} = 0.5\overline{\color{#000080}{\mathbf{123456790}}} \] \[ \frac{84}{162} = 0.5\overline{\color{#000080}{\mathbf{185}}} \] \[ \frac{85}{162} = 0.5\overline{\color{#000080}{\mathbf{246913580}}} \] \[ \frac{86}{162} = 0.5\overline{\color{#000080}{\mathbf{308641975}}} \] \[ \frac{87}{162} = 0.5\overline{\color{#000080}{\mathbf{370}}} \] \[ \frac{88}{162} = 0.5\overline{\color{#000080}{\mathbf{432098765}}} \] \[ \frac{89}{162} = 0.5\overline{\color{#000080}{\mathbf{493827160}}} \] \[ \frac{90}{162} = 0.5\overline{\color{#000080}{\mathbf{5}}} \]
\[ \frac{91}{162} = 0.5\overline{\color{#000080}{\mathbf{617283950}}} \] \[ \frac{92}{162} = 0.5\overline{\color{#000080}{\mathbf{679012345}}} \] \[ \frac{93}{162} = 0.5\overline{\color{#000080}{\mathbf{740}}} \] \[ \frac{94}{162} = 0.5\overline{\color{#000080}{\mathbf{802469135}}} \] \[ \frac{95}{162} = 0.5\overline{\color{#000080}{\mathbf{864197530}}} \] \[ \frac{96}{162} = 0.5\overline{\color{#000080}{\mathbf{925}}} \] \[ \frac{97}{162} = 0.5\overline{\color{#000080}{\mathbf{987654320}}} \] \[ \frac{98}{162} = 0.6\overline{\color{#000080}{\mathbf{049382716}}} \] \[ \frac{99}{162} = 0.6\overline{\color{#000080}{\mathbf{1}}} \]
periodo 061728395

de 1 en 1

sin 4

 periodo 123456790

de 5 en 5

sin 8

 periodos185,740,296

sin 3

periodo 246913580

de 7 en 7

sin 7

 periodo 308641975

de 2 en 2

sin 2

 periodo 370,925,481

sin 6

periodo 432098765

de 4 en 4

sin 1

 periodo 493827160

de 8 en 8

sin 5

 periodo 5,1,6,2,7,3,8,4,0

sin 9

En la primera columna al periodo le falta el 4

En la 2ª columna al periodo le falta el 8 pues 2·4=8

En la 3ª columna al periodo le falta el 3 pues 3·4=12 y 1+2=3

En la 4ª columna al periodo le falta el 7 pues 4·4=16 y 1+6=7

...

En la 9ª columna al periodo le falta el 9 pues 9·4=36 y 3+6=9

n/81

Obsérvense los periodos en cada columna

En la última fila vemos algunas observaciones

\[ \frac{1}{81} = 0.\overline{\color{#000080}{\mathbf{012345679}}} \] \[ \frac{2}{81} = 0.\overline{\color{#000080}{\mathbf{024691358}}} \] \[ \frac{3}{81} = 0.\overline{\color{#000080}{\mathbf{037}}} \] \[ \frac{4}{81} = 0.\overline{\color{#000080}{\mathbf{049382716}}} \] \[ \frac{5}{81} = 0.\overline{\color{#000080}{\mathbf{061728395}}} \] \[ \frac{6}{81} = 0.\overline{\color{#000080}{\mathbf{074}}} \] \[ \frac{7}{81} = 0.\overline{\color{#000080}{\mathbf{086419753}}} \] \[ \frac{8}{81} = 0.\overline{\color{#000080}{\mathbf{098765432}}} \] \[ \frac{9}{81} = 0.\overline{\color{#000080}{\mathbf{1}}} \]
\[ \frac{10}{81} = 0.\overline{\color{#000080}{\mathbf{123456790}}} \] \[ \frac{11}{81} = 0.\overline{\color{#000080}{\mathbf{135802469}}} \] \[ \frac{12}{81} = 0.\overline{\color{#000080}{\mathbf{148}}} \] \[ \frac{13}{81} = 0.\overline{\color{#000080}{\mathbf{160493827}}} \] \[ \frac{14}{81} = 0.\overline{\color{#000080}{\mathbf{172839506}}} \] \[ \frac{15}{81} = 0.\overline{\color{#000080}{\mathbf{185}}} \] \[ \frac{16}{81} = 0.\overline{\color{#000080}{\mathbf{197530864}}} \] \[ \frac{17}{81} = 0.\overline{\color{#000080}{\mathbf{209876543}}} \] \[ \frac{18}{81} = 0.\overline{\color{#000080}{\mathbf{2}}} \]
\[ \frac{19}{81} = 0.\overline{\color{#000080}{\mathbf{234567901}}} \] \[ \frac{20}{81} = 0.\overline{\color{#000080}{\mathbf{246913580}}} \] \[ \frac{21}{81} = 0.\overline{\color{#000080}{\mathbf{259}}} \] \[ \frac{22}{81} = 0.\overline{\color{#000080}{\mathbf{271604938}}} \] \[ \frac{23}{81} = 0.\overline{\color{#000080}{\mathbf{283950617}}} \] \[ \frac{24}{81} = 0.\overline{\color{#000080}{\mathbf{296}}} \] \[ \frac{25}{81} = 0.\overline{\color{#000080}{\mathbf{308641975}}} \] \[ \frac{26}{81} = 0.\overline{\color{#000080}{\mathbf{320987654}}} \] \[ \frac{27}{81} = 0.\overline{\color{#000080}{\mathbf{3}}} \]
\[ \frac{28}{81} = 0.\overline{\color{#000080}{\mathbf{345679012}}} \] \[ \frac{29}{81} = 0.\overline{\color{#000080}{\mathbf{358024691}}} \] \[ \frac{30}{81} = 0.\overline{\color{#000080}{\mathbf{370}}} \] \[ \frac{31}{81} = 0.\overline{\color{#000080}{\mathbf{382716049}}} \] \[ \frac{32}{81} = 0.\overline{\color{#000080}{\mathbf{395061728}}} \] \[ \frac{33}{81} = 0.\overline{\color{#000080}{\mathbf{407}}} \] \[ \frac{34}{81} = 0.\overline{\color{#000080}{\mathbf{419753086}}} \] \[ \frac{35}{81} = 0.\overline{\color{#000080}{\mathbf{432098765}}} \] \[ \frac{36}{81} = 0.\overline{\color{#000080}{\mathbf{4}}} \]
\[ \frac{37}{81} = 0.\overline{\color{#000080}{\mathbf{456790123}}} \] \[ \frac{38}{81} = 0.\overline{\color{#000080}{\mathbf{469135802}}} \] \[ \frac{39}{81} = 0.\overline{\color{#000080}{\mathbf{481}}} \] \[ \frac{40}{81} = 0.\overline{\color{#000080}{\mathbf{493827160}}} \] \[ \frac{41}{81} = 0.\overline{\color{#000080}{\mathbf{506172839}}} \] \[ \frac{42}{81} = 0.\overline{\color{#000080}{\mathbf{518}}} \] \[ \frac{43}{81} = 0.\overline{\color{#000080}{\mathbf{530864197}}} \] \[ \frac{44}{81} = 0.\overline{\color{#000080}{\mathbf{543209876}}} \] \[ \frac{45}{81} = 0.\overline{\color{#000080}{\mathbf{5}}} \]
\[ \frac{46}{81} = 0.\overline{\color{#000080}{\mathbf{567901234}}} \] \[ \frac{47}{81} = 0.\overline{\color{#000080}{\mathbf{580246913}}} \] \[ \frac{48}{81} = 0.\overline{\color{#000080}{\mathbf{592}}} \] \[ \frac{49}{81} = 0.\overline{\color{#000080}{\mathbf{604938271}}} \] \[ \frac{50}{81} = 0.\overline{\color{#000080}{\mathbf{617283950}}} \] \[ \frac{51}{81} = 0.\overline{\color{#000080}{\mathbf{629}}} \] \[ \frac{52}{81} = 0.\overline{\color{#000080}{\mathbf{641975308}}} \] \[ \frac{53}{81} = 0.\overline{\color{#000080}{\mathbf{654320987}}} \] \[ \frac{54}{81} = 0.\overline{\color{#000080}{\mathbf{6}}} \]
\[ \frac{55}{81} = 0.\overline{\color{#000080}{\mathbf{679012345}}} \] \[ \frac{56}{81} = 0.\overline{\color{#000080}{\mathbf{691358024}}} \] \[ \frac{57}{81} = 0.\overline{\color{#000080}{\mathbf{703}}} \] \[ \frac{58}{81} = 0.\overline{\color{#000080}{\mathbf{716049382}}} \] \[ \frac{59}{81} = 0.\overline{\color{#000080}{\mathbf{728395061}}} \] \[ \frac{60}{81} = 0.\overline{\color{#000080}{\mathbf{740}}} \] \[ \frac{61}{81} = 0.\overline{\color{#000080}{\mathbf{753086419}}} \] \[ \frac{62}{81} = 0.\overline{\color{#000080}{\mathbf{765432098}}} \] \[ \frac{63}{81} = 0.\overline{\color{#000080}{\mathbf{7}}} \]
\[ \frac{64}{81} = 0.\overline{\color{#000080}{\mathbf{790123456}}} \] \[ \frac{65}{81} = 0.\overline{\color{#000080}{\mathbf{802469135}}} \] \[ \frac{66}{81} = 0.\overline{\color{#000080}{\mathbf{814}}} \] \[ \frac{67}{81} = 0.\overline{\color{#000080}{\mathbf{827160493}}} \] \[ \frac{68}{81} = 0.\overline{\color{#000080}{\mathbf{839506172}}} \] \[ \frac{69}{81} = 0.\overline{\color{#000080}{\mathbf{851}}} \] \[ \frac{70}{81} = 0.\overline{\color{#000080}{\mathbf{864197530}}} \] \[ \frac{71}{81} = 0.\overline{\color{#000080}{\mathbf{876543209}}} \] \[ \frac{72}{81} = 0.\overline{\color{#000080}{\mathbf{8}}} \]
\[ \frac{73}{81} = 0.\overline{\color{#000080}{\mathbf{901234567}}} \] \[ \frac{74}{81} = 0.\overline{\color{#000080}{\mathbf{913580246}}} \] \[ \frac{75}{81} = 0.\overline{\color{#000080}{\mathbf{925}}} \] \[ \frac{76}{81} = 0.\overline{\color{#000080}{\mathbf{938271604}}} \] \[ \frac{77}{81} = 0.\overline{\color{#000080}{\mathbf{950617283}}} \] \[ \frac{78}{81} = 0.\overline{\color{#000080}{\mathbf{962}}} \] \[ \frac{79}{81} = 0.\overline{\color{#000080}{\mathbf{975308641}}} \] \[ \frac{80}{81} = 0.\overline{\color{#000080}{\mathbf{987654320}}} \] \[ \frac{81}{81} = 1.\overline{\color{#000080}{\mathbf{0}}} \]
\[ \frac{82}{81} = 1.\overline{\color{#000080}{\mathbf{012345679}}} \] \[ \frac{83}{81} = 1.\overline{\color{#000080}{\mathbf{024691358}}} \] \[ \frac{84}{81} = 1.\overline{\color{#000080}{\mathbf{037}}} \] \[ \frac{85}{81} = 1.\overline{\color{#000080}{\mathbf{049382716}}} \] \[ \frac{86}{81} = 1.\overline{\color{#000080}{\mathbf{061728395}}} \] \[ \frac{87}{81} = 1.\overline{\color{#000080}{\mathbf{074}}} \] \[ \frac{88}{81} = 1.\overline{\color{#000080}{\mathbf{086419753}}} \] \[ \frac{89}{81} = 1.\overline{\color{#000080}{\mathbf{098765432}}} \] \[ \frac{90}{81} = 1.\overline{\color{#000080}{\mathbf{1}}} \]
\[ \frac{91}{81} = 1.\overline{\color{#000080}{\mathbf{123456790}}} \] \[ \frac{92}{81} = 1.\overline{\color{#000080}{\mathbf{135802469}}} \] \[ \frac{93}{81} = 1.\overline{\color{#000080}{\mathbf{148}}} \] \[ \frac{94}{81} = 1.\overline{\color{#000080}{\mathbf{160493827}}} \] \[ \frac{95}{81} = 1.\overline{\color{#000080}{\mathbf{172839506}}} \] \[ \frac{96}{81} = 1.\overline{\color{#000080}{\mathbf{185}}} \] \[ \frac{97}{81} = 1.\overline{\color{#000080}{\mathbf{197530864}}} \] \[ \frac{98}{81} = 1.\overline{\color{#000080}{\mathbf{209876543}}} \] \[ \frac{99}{81} = 1.\overline{\color{#000080}{\mathbf{2}}} \]
periodo 012345679

de 1 en 1

sin 8

 periodo 246913580

de 5 en 5

sin 7

 periodos 037, 148, 259

sin 6

periodo 049382716

de 7 en 7

sin 5

 periodo 061728395

de 2 en 2

sin 4

 periodos 074, 185, 296

sin 3

periodo 086419753

de 4 en 4

sin 2

 periodo 098765432

de 8 en 8

sin 1

 periodos 1,2,3,4,5,6,7,8,0

sin 9

En la primera columna al periodo le falta el 8

En la 2ª columna al periodo le falta el 7 pues 2·8=16 y 1+6=7

En la 3ª columna al periodo le falta el 6 pues 3·8=24 y 2+4=6

En la 4ª columna al periodo le falta el 5 pues 4·8=32 y 3+2=5

...

En la 9ª columna al periodo le falta el 9 pues 9·8=72 y 7+2=9

 

 

RESUMEN

n/2025
per 049382716

de 1 en 1

sin 5

 per 098765432

de 5 en 5

sin 1

 per 148,592,037

sin 6

per 197530864

de 7 en 7

sin 2

 per 024691358

de 2 en 2

sin 7

 per 296,074,185

sin 3

per 345679012

de 4 en 4

sin 8

 per 395061728

de 8 en 8

sin 4

 per 4,8,3,7,2,6,1,5,0,

sin 9
n/162
per 061728395

de 1 en 1

sin 4

 per 123456790

de 5 en 5

sin 8

 per 185,740,296

sin 3

per 246913580

de 7 en 7

sin 7

 per 308641975

de 2 en 2

sin 2

 per 370,925,481

sin 6

per 432098765

de 4 en 4

sin 1

 per 493827160

de 8 en 8

sin 5

 per 5,1,6,2,7,3,8,4,0

sin 9
n/81
per 012345679

de 1 en 1

sin 8

 per 246913580

de 5 en 5

sin 7

 per 037,148,259

sin 6

per 049382716

de 7 en 7

sin 5

 per 061728395

de 2 en 2

sin 4

 per 074,185,296

sin 3

per 086419753

de 4 en 4

sin 2

 per 098765432

de 8 en 8

sin 1

 per 1,2,3,4,5,6,7,8,0

sin 9

 

Estos periodos son siempre los mismos para números del tipo 2i5j92

pues el periodo está definido por los factores primos diferentes de 2 y 5, ya que si aquí por ejemplo, en 1/2i5j92 multiplicamos por 5i2j el denominador, solo añadimos ceros que influirán en el anteperiodo, no cambia el periodo.

La pregunta ahora es ¿cuántas cifras tiene el periodo de 1/piqj siendo p y q números primos diferentes de 2 y 5?

El periodo de n/9 tiene una cifra
El periodo de n/92 tiene 9 cifrás salvo que n sea múltiplo de 9 que tendrá una, o de 3 que tendrá 3

El periodo de n/2i es 0 pues es igual a  n·5i/10i

Análogamente se concluye que el periodo de n/(2i 5j) es 0

El periodo de n/(2i 5j m) es el mismo que el de un número/m?
 n/(2i 5j m)=n 5i 2j /(10i+j m) que tiene el mismo periodo que n 5i 2j /m

Así que estudiaremos los periodos de n/m siendo m impar y que no acaba en 5

 

el periodo de n/3 tiene una cifra

el periodo de n/32 tiene 30 cifra

el periodo de n/33 tiene 31 cifras

el periodo de n/34 tiene 32 cifras

el periodo de n/35 tiene 33 cifras

 el periodo de n/7 tiene 6 cifras

el periodo de n/72 tiene 6·71 cifras

el periodo de n/73 tiene 6·72 cifras

el periodo de n/74 tiene 6·73 cifras

el periodo de n/75 tiene 6·74 cifras

 el periodo de n/11 tiene 2 cifras

el periodo de n/112 tiene 2·111 cifras

el periodo de n/113 tiene 2·112 cifras

el periodo de n/114 tiene 2·113 cifras

el periodo de n/115 tiene 2·114 cifras

 el periodo de n/13 tiene 6 cifras

el periodo de n/132 tiene 6·131 cifras

el periodo de n/133 tiene 6·132 cifras

el periodo de n/134 tiene 6·133 cifras

el periodo de n/135 tiene 6·134 cifras